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 → B^G be an R-algebra map. Assume that A is Noetherian. We show that if the induced map K ⊗ A → (K ⊗ B)^K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then S ⊗ A → (S ⊗ B)^S⊗G is an isomorphism for any R-algebra S.