Base change of invariant subrings

Let $R$ be a Dedekind domain, $G$ an affine flat $R$-group scheme, and $B$ a flat $R$-algebra on which $G$ acts. Let $A \to B^G$ be an $R$-algebra map. Assume that $A$ is Noetherian. We show that if the induced map $K\otimes A\to (K\otimes B)^{K\otimes G}$ is an isomorphism for any algebraically closed field $K$ which is an $R$-algebra, then $S\otimes A\to (S\otimes B)^{S\otimes G}$ is an isomorphism for any $R$-algebra $S$.