I wrote up some notes on this in 2004. There have been some developments since then that I will indicate below.
Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$ Denote by $\omega_\pi$ the relative dualizing sheaf. Since $\pi$ is flat and proper, also the fiber product morphism, $$\mathbb{P}^n\times_{\text{Spec}\ R}\mathcal{X}\to \text{Spec}\ R,$$ is also flat and proper. For a flat, proper morphism, Michael Artin proved representability of the relative Hilbert functor by an algebraic space that is locally finitely presented and separated over the base. Thus, there exists a universal pair $$(\rho:I\to \text{Spec}\ R,\phi:I\times_{\text{Spec}\ R}\mathbb{P}^n \xrightarrow{\cong} I\times_{\text{Spec}\ R}\mathcal{X})$$ of a separated, locally finitely presented scheme $I$ over $\text{Spec}\ R$ and an isomorphism $\phi$ of $I$-schemes, and this is compatible with arbitrary base change of $\text{Spec}\ R$.
Question 1. Is $\rho$ surjective on points?
This can be checked after surjective base change of $\text{Spec}\ R$. Thus, without loss of generality, assume that the generic fiber $X_K$ is $K$-isomorphic to $\mathbb{P}^n_K$. Let $H\subset X_K$ be an effective Cartier divisor whose associated invertible sheaf generates the Picard group. Since $\mathcal{X}$ is smooth over $\text{Spec}\ R$, also $\mathcal{X}$ is regular. Thus the closure $\mathcal{H}$ of $H$ in $\mathcal{X}$ is a Cartier divisor that is flat over $\text{Spec}\ R$. Denote by $\mathcal{L}$ the associated invertible sheaf (the dual of the ideal sheaf of $\mathcal{H}$). The restriction of $\mathcal{L}$ to $X_K$ is an ample generator of $\text{Pic}(X_K)$.
Lemma 2. Let $\pi:\mathcal{X}\to \text{Spec}\ R$ be a proper, flat morphism of schemes, and let $\mathcal{L}$ be an invertible sheaf on $\mathcal{X}$ whose restriction to $X_K$ is a generator of the Picard group. If the closed fiber $X_k$ is integral, then every invertible sheaf on $\mathcal{X}$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$.
Proof. Since $\pi$ is proper and flat with integral closed fiber, also $\mathcal{X}$ is integral. Thus, every invertible sheaf on $\mathcal{X}$ is $R$-flat. So the pushforward by $\pi$ is a finitely generated, torsion-free $R$-module, i.e., it is a finite, free $R$-module.
For every invertible sheaf $\mathcal{M}$ on $\mathcal{X}$, since $\mathcal{L}|_{X_K}$ generates $\text{Pic}(X_K)$, there exists an integer $m$ such that $\mathcal{M}|_{X_K}$ is isomorphic to $\mathcal{L}^{\otimes m}|_{X_K}$. The pushforward $\pi_*\textit{Hom}_{\mathcal{O}_X}(\mathcal{M},\mathcal{L}^{\otimes m})$ is a finite, free $R$-module whose $K$-fiber equals $H^0(X_K,\mathcal{O}_{X_K})$. Thus, there exists an element of this $R$-module whose $K$-germ is an invertible element of $H^0(X_K,\mathcal{O}_{X_K})$. This element corresponds to a homomorphism of invertible sheaves, $$\alpha:\mathcal{M} \to \mathcal{L}^{\otimes m},$$ that is an isomorphism on $X_K$. Thus, the support of the cokernel of $\alpha$ is an effective Cartier divisor on $\mathcal{X}$ that is disjoint from $X_K$.
Since the closed fiber $X_k$ is integral, the support of the cokernel equals $r\underline{X}_k$ for some integer $r\geq 0$. Thus $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}(-r\underline{X}_k)$ for some integer $r$. Since $\mathcal{O}(-\underline{X}_k)$ is the pullback by $\pi$ of the maximal ideal of $R$, and since this is a free $R$-module, also $\mathcal{O}(-\underline{X}_k)$ is a free $\mathcal{O}_{\mathcal{X}}$-module. Therefore, $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}$. QED
Proposition 3. With the same hypotheses as in Lemma 1, assume also that $\mathcal{X}$ is regular. Then either the invertible sheaf $\mathcal{L}$ is $\pi$-ample or the invertible sheaf $\mathcal{L}^\vee$ is $\pi$-ample.
Proof. This is automatic if $\pi$ is finite. Hence assume that $\pi$ has positive fiber dimension. By Chow's Lemma, a blowing up of $X_K$ is projective. This blowing up is covered by complete intersection curves whose pushforwards to $X_K$ are numerically equivalent. Up to replacing $\mathcal{L}$ by $\mathcal{L}^\vee$, assume that the degree of $\mathcal{L}$ on these curves is nonnegative. The claim is that for every integer $m$ that is sufficient positive and divisible, the invertible sheaf $\mathcal{L}^{\otimes m}$ is globally generated and the complete linear system defines a finite morphism to projective space.
For every open affine $U \subset \mathcal{X}$ whose closed fiber $U_k$ is nonempty, for every pair $p,q\in U_k$ of distinct closed points, there exists $f\in H^0(U,\mathcal{O}_{\mathcal{X}})$ that is zero on $p$ and that is nonzero on $q$. The zero scheme $\text{Zero}(f)$ is a Cartier divisor in $U$. The closure in $\mathcal{X}$ of $\text{Zero}(f)$ is a Cartier divisor $D$ that contains $p$, yet does not contain $q$. By the previous lemma, $\mathcal{O}(\underline{D})$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$. Since $D$ is effective, its restriction to $X_K$ is effective. Thus, the intersection number with the covering family of curves is positive. Therefore, the integer $m$ is also positive. Via this isomorphism, there exists a global section $s$ of $\mathcal{L}^{\otimes m}$ that vanishes at $p$ yet does not vanish at $q$. Note that for every positive integer $d$, also $s^d$ is a global section of $\mathcal{L}^{\otimes md}$ that vanishes at $p$ yet does not vanish at $q$.
Thus, for every integer $d\geq 1$, for the base locus $B_d$ of the complete linear system of $\mathcal{L}^{\otimes d}$, for every $q\in B_d$, there exists an integer $m\geq 1$ such that $B_{dm}$ is contained in $B_d$ yet does not contain $q$. By Noetherian induction, for all integers $m$ that are sufficiently positive and divisible, $B_m$ is empty. Moreover, for every closed point $p\in X_k$, for every open affine $U$ neighborhood of $p$, for every closed point $q\in U_k$, there exists an integer $m\geq 1$ such that the associated $R$-morphism $$\phi_m:\mathcal{X}\to \mathbb{P}H^0(\mathcal{X},\mathcal{L}^{\otimes m}),$$ does not contain $(p,q)$ in the fiber product $$\mathcal{X}\times_{\mathbb{P}H^0(\mathcal{L}^m)}\mathcal{X}\subset \mathcal{X}\times_{\text{Spec}\ R}\mathcal{X}.$$ Thus, by another Noetherian induction argument, for all integers $m$ that are sufficiently positive and divisible, every irreducible component of the fiber product projects finitely to $\mathcal{X}$ under projection on the first factor. Thus, $\phi_m$ is a finite morphism, and $\mathcal{L}^{\otimes m}$ is ample. QED
Nota bene. There is a vast generalization of Lemma 2 for specializations of rationally connected varieties in the following article of Gounelas and Javanpeykar.
Frank Gounelas, Ariyan Javanpeykar
Invariants of Fano varieties in families
https://arxiv.org/abs/1703.0573
I will complete the argument soon. The basic idea now is to use specializations of lines to prove that $\mathcal{H}$ is smooth over $\text{Spec}\ R$ (or else the multiplicity at a singular point would give an intersection number with a specialization of a line that is greater than one). Then, by an induction hypothesis, the scheme $\mathcal{H}$ is a projective space bundle over $\text{Spec}\ R$. Finally, fixing a $k$-point $q$ of $X_k$ that is not contained in $H_k$, the specializations in $X_k$ of lines that contain $q$ each intersect $H_k$ in a unique point. From this, it follows that the blowing up of $X_k$ along $q$ is isomorphic to the projective bundle over $H_k$ of $\mathcal{O}_{H_k}\oplus \mathcal{L}|_{H_k}$. Using the induction hypothesis, this equals the blowing up of $\mathbb{P}^n_k$ along a $k$-point.