We offer two facts and a Theorem.
Let $S$ be a commutative noetherian ring containing $\mathbb Z$ and let $G=G_S$ be
reductive over $S$ in the sense of SGA3. That is, $G$ is smooth over $S$
with geometric fibers that are connected reductive.
Let $R$ be a finitely generated commutative $S$-algebra.
Suppose that $\mathrm{Spec}(R)$ is equipped with a right
$G$-action over $S$.
Fact 1 Let us be given a base change $S\to \bf k$ to a field of positive characteristic $p$.
For every $x\in (R\otimes_S {\bf k})^G$ there is an $r\geq1$ so that $x^{p^r}$ lies in the $\bf k$-span of the image
of $R^G$.
Fact 2 For almost all primes $p$ the map $R^G\to (R/pR)^G$ is surjective.
Remark. We do not need to distinguish between $(R/pR)^G$ and $(R/pR)^{G_{S/pS}}$. Base change of the group
is unnecessary when computing invariants or cohomology. (See 1.13. Restriction in our survey
Reductivity properties over an affine base
.)
Theorem Assume further that $R$ is flat over $S$ and that $S$ is of finite global homological dimension.
Then there is an integer $n\geq1$ so that if $S[1/n]\to \bf k$ is a base change to a field, then the map
$R^G\otimes_S{\bf k}\to (R\otimes_S{\bf k})^{G_{\bf k}}$ is surjective.
To prove Fact 1, let $D$ be the image of $S$ in $\bf k$. As $D\to \bf k$ is flat, we have $(R\otimes_S {\bf k})^G=(R\otimes_S D)^G\otimes_D {\bf k}$,
so it suffices to show that for every $x\in (R\otimes_S D)^G$ there is an $r\geq1$ so that $x^{p^r}$ lies in the the image
of $R^G$. Now $G$ is power reductive, so we may apply Proposition 41
of our paper
Power Reductivity over an Arbitrary Base
.
See also my survey
Reductivity properties over an affine base
.
To prove Fact 2, recall from Theorem 10.5 of
Good Grosshans filtration in a family
that $H^1(G,R)$ is a finitely generated module over $R^G$.
This module is also $\mathbb Z$-torsion.
To see this, first take an fppf base change to reduce to the case that $G$ is split over $S$ (see SGA3).
Then $G_{\mathbb Q}$ makes sense and
$H^1(G,R)\otimes_{\mathbb Z}{\mathbb Q}=H^1(G_{\mathbb Q},R\otimes_{\mathbb Z}{\mathbb Q})=0$.
Choose $n\geq1$ so that $n$ annihilates the generators of $H^1(G,R)$. Choose $m\geq1$ so that $m$
annihilates the generators of the $\mathbb Z$-torsion ideal of $R$.
Now if $p$ does not divide $mn$, then $\partial$ vanishes in the exact sequence
$0\to R^G\stackrel{\times p}\to R^G\to (R/pR)^G\stackrel\partial\to H^1(G,R)$.
Now we turn to the proof of the Theorem. If $G$ is split over $S$ then we may apply
Remark 31 and Theorem 33 of
Power Reductivity over an Arbitrary Base
to obtain $n$ so that $H^i(G,R[1/n])$ vanishes for $i\geq1$.
If $G$ is not yet split the same result is true. Indeed we may by SGA3 do an fppf base change $S\to T$ so that $G_T$ is split over $T$.
Then we may arrange that $H^i(G,R[1/n])\otimes_ST=H^i(G_T,R[1/n]\otimes_ST)$ vanishes for $i\geq1$. This implies that $H^i(G,R[1/n])$ vanishes for $i\geq1$.
Having chosen $n$ this way we now claim that for every $S$-module $N$ with trivial $G$ action, the
$H^i(G,R[1/n]\otimes_SN)$ also vanish for $i\geq1$. This is clear if $N$ is free and then it follows by induction on the projective dimension of the $S$-module $N$.
(If $0\to N'\to F \to N\to0$ is exact, with $F$ free, consider the long exact sequence for $G$-cohomology associated with the exact sequence
$0\to R[1/n]\otimes_SN'\to R[1/n]\otimes_SF \to R[1/n]\otimes_SN\to0$.)
Now let us be given a base change $S[1/n]\to \bf k$ to a field. Let $N$ be the kernel of $S[1/n]\to \bf k$ and let $D$ be its image.
Note that $0\to R\otimes_SN\to R\otimes_SS[1/n] \to R\otimes_SD\to0$ is exact and that $R\otimes_SN=R[1/n]\otimes_SN$.
As $H^1(G,R[1/n]\otimes_SN)=0$ we have a surjection $(R\otimes_SS[1/n])^G\to (R\otimes_SD)^G$.
As $D\to \bf k$ is flat, $(R\otimes_S{\bf k})^{G_{\bf k}}=(R\otimes_SD)^G\otimes_D{\bf k}$.
We see that $(R\otimes_SS[1/n])^G\otimes_S{\bf k}$ maps onto $(R\otimes_S{\bf k})^{G_{\bf k}}$. But $(R\otimes_SS[1/n])^G\otimes_S{\bf k}$ equals
$R^G\otimes_SS[1/n]\otimes_S{\bf k}=R^G\otimes_S{\bf k}$. The result follows.
