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Edit. Fixed some mistakes about Zariski tangent spaces. For all of the torsion, cyclic cotangent sheaves $\Omega_u$ with support $R$, it is necesssary to pass to the $u$-relative dual sheaf $\Omega^\vee_u := \textit{Hom}_{\mathcal{O}_C}(\Omega_u\otimes \omega_u, \mathcal{O}_R)$, which is isomorphic to $\Omega_u$ as a torsion, cyclic coherent sheaf on $C$, yet is better adapted to infinitesimal deformation theory.

By hypothesis, each $\lambda_{i,j}$ is an integer $\geq 2$ that is prime to $p$. Thus, the sheaf of relative differentials, $\Omega_u$, is a cyclic, torsion $\mathcal{O}_C$-module supported on the points $r_{i,j}$. For each $(i,j)$, denote by $\Omega_{i,j}$ the stalk of $\Omega_u$ at $r_{i,j}$. Denote by $\mathfrak{m}$ the maximal ideal of $r_{i,j}$. Then $\Omega_{i,j}$ is cyclic of length $\lambda_{i,j}-1.$ In particular, $\mathfrak{m}^{\lambda_{i,j}-1}\Omega_{i,j}$ is zero, but $\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j}$ is nonzero. The $u$-relative dual of $\Omega_u$ is the following sheaf, $$\Omega_u^\vee := \textit{Hom}_{\mathcal{O}_C}(\Omega_u\otimes \omega_u,\mathcal{O}_R).$$ This is isomorphic to $\Omega_u$ as a $\mathcal{O}_C$-module, but $\Omega_u^\vee$ is slightly better for deformation theory. As above, denote by $\Omega_{i,j}^\vee$ the stalk at $r_{i,j}$ of the torsion, cyclic sheaf $\Omega_u^\vee \cong \Omega_u$.

Edit. Fixed some mistakes about Zariski tangent spaces. For all of the torsion, cyclic cotangent sheaves $\Omega_u$ with support $R$, it is necesssary to pass to the $u$-relative dual sheaf $\Omega^\vee_u := \textit{Hom}_{\mathcal{O}_C}(\Omega_u, \omega_u\otimes \mathcal{O}_R)$, which is isomorphic to $\Omega_u$ as a torsion, cyclic coherent sheaf on $C$, yet is better adapted to infinitesimal deformation theory.

By hypothesis, each $\lambda_{i,j}$ is an integer $\geq 2$ that is prime to $p$. Thus, the sheaf of relative differentials, $\Omega_u$, is a cyclic, torsion $\mathcal{O}_C$-module supported on the points $r_{i,j}$. For each $(i,j)$, denote by $\Omega_{i,j}$ the stalk of $\Omega_u$ at $r_{i,j}$. Denote by $\mathfrak{m}$ the maximal ideal of $r_{i,j}$. Then $\Omega_{i,j}$ is cyclic of length $\lambda_{i,j}-1.$ In particular, $\mathfrak{m}^{\lambda_{i,j}-1}\Omega_{i,j}$ is zero, but $\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j}$ is nonzero. The $u$-relative dual of $\Omega_u$ is the following sheaf, $$\Omega_u^\vee := \textit{Hom}_{\mathcal{O}_C}(\Omega_u,\omega_u\otimes\mathcal{O}_R).$$ This is isomorphic to $\Omega_u$ as a $\mathcal{O}_C$-module, but $\Omega_u^\vee$ is slightly better for deformation theory. As above, denote by $\Omega_{i,j}^\vee$ the stalk at $r_{i,j}$ of the torsion, cyclic sheaf $\Omega_u^\vee \cong \Omega_u$.

I do not know a reference. The proof is straightforward, at least when the characteristic $p$ equals $0$ or is $>d$.

Splitting ramification. For every finite and flat morphism with Noetherian source and target, $$f:S\to T,$$ consider the coherent $\mathcal{O}_S$-module of relative differentials $\Omega_f$. This is $f$-finite, but it typically is not $f$-flat.

Proof. This can be checked after base change. Thus, assume that $k$ is algebraically closed. Let $$u:C\to \mathbb{P}^1_k,$$ be a given degree $d$ $k$-morphism with $C$ a smooth, connected, projective $k$-curve of genus $g$. Let the branch points in $\mathbb{P}^1_k$ be the pairwise distinct $k$-points $(b_1,\dots,b_n)$ of $\mathbb{P}^1_k$. For each branch point, let the ramification points above $b_i$ be pairwise distinct $k$-points $(r_{i,1},\dots,r_{i,\ell_i})$ of $C$. For each $j=1,\dots,\ell_i$, let the length of $u^{-1}(b_i)$ at $r_{i,j}$ equal $\lambda_{i,j}.$ By hypothesis, each $\lambda_{i,j}$ is an integer $\geq 2$ that is prime to $p$. Thus, $\Omega_u$ is a torsion $\mathcal{O}_C$-module supported on the points $r_{i,j}$. For each $(i,j)$, denote by $\Omega_{i,j}$ the stalk of $\Omega_u$ at $r_{i,j}$. Denote by $\mathfrak{m}$ the maximal ideal of $r_{i,j}$. Then $\Omega_{i,j}$ is cyclic of length $\lambda_{i,j}-1.$ In particular, $\mathfrak{m}^{\lambda_{i,j}-1}\Omega_{i,j}$ is zero, but $\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j}$ is nonzero.

the $k$-vector space of first-order deformations of $u$ as a cover of $\mathbb{P}^1_k$ is canonically isomorphic to the $k$-vector space $H^0(C,\Omega_u)$. This, in turn, is the direct sum over all $1\leq i\leq n$ and all $1\leq j \leq \ell_i$ of the stalk $\Omega_{i,j}$. Inside this $b$-dimensional $k$-vector space, the Zariski tangent space to the component of $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho}$ is the direct sum of the submodules $\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j}$. This has dimension $\sum_i \ell_i$ as a $k$-vector space.

By relative duality for finite flat maps, there is a trace map of locally free $\mathcal{O}_{\mathbb{P}^1}$-modules, $$\text{Tr}_u:u_*\Omega_{C/k} \cong \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(u_*\mathcal{O}_C,\Omega_{\mathbb{P}^1_k/k}) \to \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{O}_{\mathbb{P}^1_k},\Omega_{\mathbb{P}^1_k/k}) = \Omega_{\mathbb{P}^1_k/k}.$$ The restriction of the trace map gives a $k$-linear isomorphism, $$\text{Tr}_{u,(i,j)}:\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ Here $\mathfrak{n}$ is the maximal ideal at $b_i$. These $k$-linear isomorphisms define a direct sum isomorphism, $$\bigoplus_{j=1}^{\ell_i} \mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \bigoplus_{j=1}^{\ell_i} \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ The inverse image of the diagonal copy in the target is a $1$-dimensional $k$-subspace, $\Omega_i$. The Zariski tangent space to $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ at $[u]$ is the direct sum over $i=1,\dots,n$ of the $1$-dimensional $k$-subspace $\Omega_i$. This is an $n$-dimensional $k$-vector space. Thus, $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ is smooth of dimension $n$. Moreover, the derivative of the branch map is the $k$-isomorphism induced by the trace maps, $$\bigoplus_{i=1}^n \Omega_i \to \bigoplus_{i=1}^n \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ Thus, the branch map is étale. QED

Edit. Fixed some mistakes about Zariski tangent spaces. For all of the torsion, cyclic cotangent sheaves $\Omega_u$ with support $R$, it is necesssary to pass to the $u$-relative dual sheaf $\Omega^\vee_u := \textit{Hom}_{\mathcal{O}_C}(\Omega_u\otimes \omega_u, \mathcal{O}_R)$, which is isomorphic to $\Omega_u$ as a torsion, cyclic coherent sheaf on $C$, yet is better adapted to infinitesimal deformation theory.

I do not know a reference. The proof is straightforward, at least when the characteristic $p$ equals $0$ or is $>d$.

Splitting tame covers. For every finite and flat morphism with Noetherian source and target, $$f:S\to T,$$ consider the coherent $\mathcal{O}_S$-module of relative differentials $\Omega_f$. This is $f$-finite, but it typically is not $f$-flat.

Proof. This can be checked after base change. Thus, assume that $k$ is algebraically closed. Let $$u:C\to \mathbb{P}^1_k,$$ be a given degree $d$ $k$-morphism with $C$ a smooth, connected, projective $k$-curve of genus $g$. Let the branch points in $\mathbb{P}^1_k$ be the pairwise distinct $k$-points $(b_1,\dots,b_n)$ of $\mathbb{P}^1_k$. Denote the full branch divisor by $$B:=\sum_i \underline{b}_i.$$

For each branch point, let the ramification points above $b_i$ be pairwise distinct $k$-points $(r_{i,1},\dots,r_{i,\ell_i})$ of $C$. For each $j=1,\dots,\ell_i$, let the length of $u^{-1}(b_i)$ at $r_{i,j}$ equal $\lambda_{i,j}.$ Denote by $R_u$ the ramification divisor $$R_u=\sum_{i,j}\lambda_{i,j}\underline{r}_{i,j}.$$

By hypothesis, each $\lambda_{i,j}$ is an integer $\geq 2$ that is prime to $p$. Thus, the sheaf of relative differentials, $\Omega_u$, is a cyclic, torsion $\mathcal{O}_C$-module supported on the points $r_{i,j}$. For each $(i,j)$, denote by $\Omega_{i,j}$ the stalk of $\Omega_u$ at $r_{i,j}$. Denote by $\mathfrak{m}$ the maximal ideal of $r_{i,j}$. Then $\Omega_{i,j}$ is cyclic of length $\lambda_{i,j}-1.$ In particular, $\mathfrak{m}^{\lambda_{i,j}-1}\Omega_{i,j}$ is zero, but $\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j}$ is nonzero. The $u$-relative dual of $\Omega_u$ is the following sheaf, $$\Omega_u^\vee := \textit{Hom}_{\mathcal{O}_C}(\Omega_u\otimes \omega_u,\mathcal{O}_R).$$ This is isomorphic to $\Omega_u$ as a $\mathcal{O}_C$-module, but $\Omega_u^\vee$ is slightly better for deformation theory. As above, denote by $\Omega_{i,j}^\vee$ the stalk at $r_{i,j}$ of the torsion, cyclic sheaf $\Omega_u^\vee \cong \Omega_u$.

the $k$-vector space of first-order deformations of $u$ as a cover of $\mathbb{P}^1_k$ is canonically isomorphic to the $k$-vector space $$H^0(C,\Omega_u^\vee).$$ This, in turn, is the direct sum over all $1\leq i\leq n$ and all $1\leq j \leq \ell_i$ of the stalk $\Omega^\vee_{i,j}$. Inside this $b$-dimensional $k$-vector space, the Zariski tangent space to the component of $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho}$ is the direct sum of the submodules $\mathfrak{m}^{\lambda_{i,j}-2}\Omega^\vee_{i,j}$. This has dimension $\sum_i \ell_i$ as a $k$-vector space.

By relative duality for finite flat maps, there is a trace map of locally free $\mathcal{O}_{\mathbb{P}^1}$-modules, $$\text{Tr}_u:u_*\Omega_{C/k} \cong \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(u_*\mathcal{O}_C,\Omega_{\mathbb{P}^1_k/k}) \to \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{O}_{\mathbb{P}^1_k},\Omega_{\mathbb{P}^1_k/k}) = \Omega_{\mathbb{P}^1_k/k}.$$ For the $u$-relative dual, this induces a trace map, $$\text{Tr}_u:u_*\Omega_u^\vee \to \Omega_{\mathbb{P}^1_k/k}^\vee|_B.$$ The restriction of the trace map gives a $k$-linear isomorphism, $$\text{Tr}_{u,(i,j)}:\mathfrak{m}^{\lambda_{i,j}-2}\Omega^\vee_{i,j} \to \Omega^\vee_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega^\vee_{\mathbb{P}^1_k/k,b_i}.$$ Here $\mathfrak{n}$ is the maximal ideal at $b_i$. These $k$-linear isomorphisms define a direct sum isomorphism, $$\bigoplus_{j=1}^{\ell_i} \mathfrak{m}^{\lambda_{i,j}-2}\Omega^\vee_{i,j} \to \bigoplus_{j=1}^{\ell_i} \Omega^\vee_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega^\vee_{\mathbb{P}^1_k/k,b_i}.$$ The inverse image of the diagonal copy in the target is a $1$-dimensional $k$-subspace, $\Omega^\vee_i$. The Zariski tangent space to $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ at $[u]$ is the direct sum over $i=1,\dots,n$ of the $1$-dimensional $k$-subspace $\Omega^\vee_i$. This is an $n$-dimensional $k$-vector space. Thus, $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ is smooth of dimension $n$. Moreover, the derivative of the branch map is the $k$-isomorphism induced by the trace maps, $$\bigoplus_{i=1}^n \Omega^\vee_i \to \bigoplus_{i=1}^n \Omega^\vee_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega^\vee_{\mathbb{P}^1_k/k,b_i}.$$ Thus, the branch map is étale. QED

Hypothesis 2. The morphism $f$ is curvilinear and tame, i.e., every connected component of every fiber of $f$ over every geometric point of $T$, say $\text{Spec}\ k \to T$, is of the form $\text{Spec}\ k[s]/\langle s^{m+1}\rangle$ for an integer $m\geq 0$ that is prime to $p$.

By relative duality for finite flat maps, there is a trace map of locally free $\mathcal{O}_{\mathbb{P}^1}$-modules, $$\text{Tr}_u:u_*\Omega_{C/k} \cong \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(u_*\mathcal{O}_C,\Omega_{\mathbb{P}^1_k/k}) \to \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{O}_{\mathbb{P}^1_k},\Omega_{\mathbb{P}^1_k/k}) = \Omega_{\mathbb{P}^1_k/k}.$$ The restriction of the trace map gives a $k$-linear isomorphism, $$\text{Tr}_{u,(i,j)}:\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbf{P}^1_k/k,b_i}.$$ Here $\mathfrak{n}$ is the maximal ideal at $b_i$. These $k$-linear isomorphisms define a direct sum isomorphism, $$\bigoplus_{j=1}^{\ell_i} \mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \bigoplus_{j=1}^{\ell_i} \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbf{P}^1_k/k,b_i}.$$ The inverse image of the diagonal copy in the target is a $1$-dimensional $k$-subspace, $\Omega_i$. The Zariski tangent space to $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ at $[u]$ is the direct sum over $i=1,\dots,n$ of the $1$-dimensional $k$-subspace $\Omega_i$. This is an $n$-dimensional $k$-vector space. Thus, $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ is smooth of dimension $n$. Moreover, the derivative of the branch map is the $k$-isomorphism induced by the trace maps, $$\bigoplus_{i=1}^n \Omega_i \to \bigoplus_{i=1}^n \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbf{P}^1_k/k,b_i}.$$ Thus, the branch map is étale. QED

Hypothesis 2. The morphism $f$ is curvilinear and tame, i.e., every connected component of every fiber of $f$ over every geometric point of $T$, say $\text{Spec}\ k \to T$, is of the form $\text{Spec}\ k[s]/\langle s^{m}\rangle$ for an integer $m\geq 0$ that is prime to $p$.

By relative duality for finite flat maps, there is a trace map of locally free $\mathcal{O}_{\mathbb{P}^1}$-modules, $$\text{Tr}_u:u_*\Omega_{C/k} \cong \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(u_*\mathcal{O}_C,\Omega_{\mathbb{P}^1_k/k}) \to \textit{Hom}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{O}_{\mathbb{P}^1_k},\Omega_{\mathbb{P}^1_k/k}) = \Omega_{\mathbb{P}^1_k/k}.$$ The restriction of the trace map gives a $k$-linear isomorphism, $$\text{Tr}_{u,(i,j)}:\mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ Here $\mathfrak{n}$ is the maximal ideal at $b_i$. These $k$-linear isomorphisms define a direct sum isomorphism, $$\bigoplus_{j=1}^{\ell_i} \mathfrak{m}^{\lambda_{i,j}-2}\Omega_{i,j} \to \bigoplus_{j=1}^{\ell_i} \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ The inverse image of the diagonal copy in the target is a $1$-dimensional $k$-subspace, $\Omega_i$. The Zariski tangent space to $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ at $[u]$ is the direct sum over $i=1,\dots,n$ of the $1$-dimensional $k$-subspace $\Omega_i$. This is an $n$-dimensional $k$-vector space. Thus, $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{n,\underline{\lambda}}$ is smooth of dimension $n$. Moreover, the derivative of the branch map is the $k$-isomorphism induced by the trace maps, $$\bigoplus_{i=1}^n \Omega_i \to \bigoplus_{i=1}^n \Omega_{\mathbb{P}^1_k/k,b_i}/\mathfrak{n}\Omega_{\mathbb{P}^1_k/k,b_i}.$$ Thus, the branch map is étale. QED