On invariants of a set of elements of a semisimple Lie algebra

Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n,N_G(h):h^n$. We study a relation between the algebra $C[h^n{]}^{N_G(h)}$ and its subalgebra consisting of restrictions to $h^n$ of elements of $C[g^n{]}^G$.