Let $H \subset G$ be an inclusion of reductive groups over an algebraically closed field $k$ of char $0$. For simplicity, let's assume that $G$ is split and $H$ contains a maximal torus for $G$. Then the map $k[G]^G \rightarrow k[H]^H$, where invariants are for the conjugation action, is injective. As is well known, it coincides with the map of representation rings induced by restriction, $R(G) \rightarrow R(H)$.
Now suppose I'm interested in $k[G^n]^G \rightarrow k[H^n]^H$, where the $G$-action on $G^n$ is via diagonal conjugation, and similarly for $H$. This should certainly not be injective for all $n$. What can one say about the kernel? In particular, is there a representation-theoretic description?
For example, one can give algebra generators for $k[G^n]^G$ in terms of representations, as of matrix coefficients for $n$-fold tensor powers of irreducible representations for $G$. Can one describe the kernel as an ideal in terms of these generators in some clean way?