The general technique goes as follows. Let $\{A_i\}$ be a directed system of rings with limit $A$; e.g., $A=k$ and $\{A_i\}$ the set of finitely generated $k_0$-subalgebras of $k$. Let $X$ and $Y$ be schemes of finite presentation over some $A_{i_0}$, and define $X_i = X \otimes_{A_{i_0}} A_i$ for $i \ge i_0$, $X_A = X \otimes_{A_{i_0}} A$, and similarly for $Y_i$ (with $i \ge i_0$) and $Y_A$. There is a natural map of sets
$$\varinjlim {\rm{Hom}}_{A_i}(X_i,Y_i) \rightarrow {\rm{Hom}}_A(X_A,Y_A)$$
and it is bijective by EGA IV$_3$, 8.8.2(i).
So given an $A$-morphism $f:X_A \rightarrow Y_A$, there exists some $i_1$ and an $A_{i_1}$-morphism $f_{i_1}:X_{i_1} \rightarrow Y_{i_1}$ which descends $f$. But we want more: for various properties $\mathbf{P}$ of morphisms of schemes, if $f$ satisfies $\mathbf{P}$ then we want $f_i := f_{i_1} \otimes_{A_{i_1}} A_i$ to also satisfy $\mathbf{P}$ for some $i \ge i_1$. The property "isomorphism" is immediate by applying the preceding formalism to the inverse $A$-morphism too (using bijectivity in the displayed map of sets above). See IV$_3$ 8.10.5 for the tip of the iceberg on many possibilities for $\mathbf{P}$. By IV$_3$, 8.6.3, open subschemes of $X_A$ with finitely presented complement also descend to such open subschemes of some $X_i$.
For $i \ge i_1$, the (set-theoretic) image
$Z_i = f_i(X_i) \subset Y_i$ is constructible and if $j \ge i$ then $Z_j$ is the preimage of $Z_i$ under $Y_j \rightarrow Y_i$ and likewise $f(X_A)$ is the preimage of $Z_i$ in $Y_A$. Define $E_i$ to be the set of of $s \in S_i := {\rm{Spec}}(A_i)$ such that $(Z_i)_s$ is dense in $(X_i)_s$,
and likewise for an analogous subset $E \subset S := {\rm{Spec}}(A)$. The density or not of a constructible subset of a scheme of finite type over a field is insensitive to extension of the ground field, so for $j \ge i$ the preimage of $E_i$ under $S_j \rightarrow S_i$ is $E_j$ (due to the analogue for $Z_i \subset Y_i$ and $Z_j \subset Y_i$ under $Y_i \rightarrow Y_i$ noted above) and likewise the preimage of $E_i$ under $S \rightarrow S_i$ is $E$. But $E_i$ is constructible in $S_i$ by IV$_3$ 9.5.3 and likewise for $E$ inside $S$. Thus, if $E=S$ then $E_i = S_i$ for all large $i$ due to IV$_3$ 8.3.5.
Thus, in the initial setup of interest, if $X_k \rightarrow Y_k$ is dominant then we get a finitely generated $k_0$-subalgebra $R \subset k$ and an $R$-morphism $h:X_R \rightarrow Y_R$ which is dominant between fibers over all points of Spec($R$).
In a similar manner, using the "spreading out" for open subschemes (with finitely presented complement) mentioned above, if there are dense open subschemes of $X_k$ and $Y_k$ between which $f$ restricts to an isomorphism, then by increasing $R$ further we can arrange that there exist fiberwise dense open subschemes $U \subset X_R$ and $V \subset Y_R$ between which $h$ restricts to an isomorphism.
Finally, thanks to the Nullstellensatz over $k_0$, using the pullback of $h$ between fibers over any closed point of such a Spec($R$) allows us to conclude for the properties of dominance, isomorphism, or birational morphism.
This overall method is sometimes called the "Principle of finite extensions" (i.e., whatever happens after some extension of the ground field already happens over a finite extension), and complete proofs of virtually every such property you could ever imagine wanting is rigorously documented in remarkable and useful generality in EGA IV$_3$, $\S8$-$\S11$ (skip $\S10$) and IV$_4$, $\S17$-$\S18$.