I do not know a reference. The proof is straightforward, at least when the characteristic $p$ equals $0$ or is $>d$.
Counterexample in the "wild" case. When the characteristic $p$ divides one of the integers $\lambda_i$, the result is false, because of moduli of ramification. For instance, consider the $1$-parameter family of morphisms of degree $d=p+1$ from the projective line to itself, $$u_t:\mathbb{P}^1_k \to \mathbb{P}^1_k, \ \ u_t([x,y]) = [x^{p+1},xy^p+ty^{p+1}], \ t\in k^\times.$$ This is a varying family in $t$, but the ramification profile and image divisor in the target $\mathbb{P}^1_k$ is fixed. Thus, assume that $p$ equals $0$ or that $p>d$. Under this hypothesis, every finite cover $f:C\to \mathbb{P}^1_k$ is tamely ramified. The $k$-stack of stable maps to $\mathbb{P}^1_k$ is a proper, finitely presented Artin $k$-stack with finite diagonal. Inside this Artin $k$-stack, there is a maximal open substack that is a Deligne-Mumford $k$-stack, $$\overline{\mathcal{M}}_{g,0}(\mathbb{P}^1_k/k,d)^{\text{DM}} \subset \overline{\mathcal{M}}_{g,0}(\mathbb{P}^1_k/k,d).$$ The hypothesis implies that the open substack $\overline{\mathcal{M}}_{g,0}(\mathbb{P}^1_k/k,d)^{\text{DM}}$ equals the entire stack $\overline{\mathcal{M}}_{g,0}(\mathbb{P}^1_k/k,d)$. In particular, this open substack contains $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)$, the locus of stable maps with smooth domain.
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.
Definition 1. The $f$-splitting stratification is the flattening stratification of $\Omega_f$. This is a locally closed immersion that is bijective on field-valued points, $$\tau_f:T_f\hookrightarrow T,$$ such that for every morphism $u:U\to T$, there is a factorization $u=\tau_f\circ \upsilon$ for a morphism $\upsilon:U\to T_f$ if and only if the pullback of $\Omega_f$ is flat over $U$.
Hypothesis 2. The morphism $f$ is curvilinear, 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 $m\geq 0$.
Notation 3. The fiber product of $f$ and $\tau_f$ is denote $S_f$, and the projection morphism is denoted $\widetilde{f}:S_f\to T_f$. The pullback of $\Omega_f$ to $S_f$ is denoted $\widetilde{\Omega}_f$. The annihilator of $\widetilde{\Omega}_f$ as an ideal sheaf in $S_f$ is denote $\widetilde{\mathcal{I}}_f$. The annihilator of the ideal sheaf $\widetilde{\mathcal{I}}_f$ as an ideal sheaf in $S_f$ is denoted $\mathcal{I}_f$. The associated closed immersion is denoted $\sigma_f:S^0_f\hookrightarrow S_f$.
Under the hypotheses, $S^0_f$ is finite and étale over $T_f$, and $\sigma_f$ is bijective on field-valued points.
Definition 4. When $f$ is curvilinear, the commutative diagram, $$\begin{array}{ccc}S_f^0 & \hookrightarrow & S \\ \downarrow & & \downarrow \\ T_f & \hookrightarrow & T \end{array},$$ is a splitting of the ramification of $f$.
Stable maps. Denote the universal stable map as follows, $$(\pi,u):\mathcal{C}\to \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d) \times_{\text{Spec}\ k}\mathbb{P}^1_k.$$ Denote by $\Omega$ the associated relative differentials of $(\pi,u)$ considered as a coherent sheaf on $\mathcal{C}$. Again by the hypothesis that $p>d$, the sheaf $\Omega$ is finite relative to $\pi$, i.e., the support of $\Omega$ maps finitely to $\mathcal{M}_{g,0}$ under $\pi$. In fact, $\Omega$ is $\pi$-flat and locally free of rank $b=2g+2d-2$. As a coherent sheaf on $\mathcal{C}$, the sheaf $\Omega$ is cyclic. Denote by $\mathcal{J}$ the annihilator ideal of $\Omega$ as an ideal sheaf on $\mathcal{C}$, and denote by $\mathcal{R}\subset \mathcal{C}$ the closed immersion associated to $\mathcal{J}$. Thus, $\Omega$ is the pushforward from $\mathcal{R}$ of an invertible sheaf. Denote by $\rho$ the restriction of $\pi$ to $\mathcal{R}$, $$\rho:\mathcal{R}\to \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d).$$ This is a finite and flat morphism of degree $b=2g+2d-2$.
As in Definition 4, denote a splitting of the ramification of $\rho$ as follows, $$ \begin{array}{ccc} \mathcal{R}^0_\rho & \xrightarrow{i} & \mathcal{R} \\ \widetilde{\rho} \downarrow & & \downarrow \rho \\ \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_\rho & \xrightarrow{\tau_\rho} & \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d) \end{array}. $$
This is not quite enough. The composition of $i$ with the universal stable map defines a finite morphism, $$(\widetilde{\rho},\widetilde{i}):\mathcal{R}^0_\rho\to \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_\rho \times_{\text{Spec}\ k}\mathbb{P}^1_k.$$ Denote the image closed immersion by $$(\varpi,\iota):\mathcal{B}_\rho\hookrightarrow \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_\rho \times_{\text{Spec}\ k}\mathbb{P}^1_k.$$ The combination of the flattening stratification of the finite morphism $\varpi$ and the splitting of the ramification of $\varpi$ gives a commutative diagram, $$ \begin{array} \mathcal{B}^0_{\rho,\varpi} & \xrightarrow{j} & \mathcal{B}_\rho \\ \widetilde{\varpi} \downarrow & & \downarrow \varpi \\ \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho,\varpi} & \xrightarrow{\tau_{\varpi}} & \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho} \end{array}. $$ The composite locally closed immersion, $$\tau_{\rho,\varpi}:\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho,\varpi} \to \mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d),$$ is a bijection on field-valued points. The pullback of the universal stable map via this base change has both split ramification locus in the domain curve and split branch locus in the domain curve. Thus, on each connected component of $\mathcal{M}_{g,0}(\mathbb{P}^1_k/k,d)_{\rho,\varpi}$, there is an associated integer $n$ and an unordered $n$-tuple of partitions of $d$, say $\{ \lambda_1,\dots, \lambda_n\}$ with lengths $\ell_i =\ell(\lambda_i)$ such that $b$ equals the sum of the weights $$w_i = w(\lambda_i) = d -\ell(\lambda_i) = d-\ell_i.$$
To be continued ...