There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by traces of monomials in the $A_i$'s (i.e. it is generated by the various $\mathrm{tr}(A_{i_1} A_{i_2} \cdots A_{i_k})$ ranging over all monomials). 

When $m=1$, this is the familiar fact that $\mathrm{tr}(A)$, $\mathrm{tr}(A^2)$, $\mathrm{tr}(A^3)$, $\dots$, $\mathrm{tr}(A^n)$ generate the ring of conjugation invariant polynommials on $\mathrm{Mat}_n$.

But in the $m=1$ case there are other very familiar choices of generating sets, corresponding to the different generating sets of the ring of symmetric polynomials. For example, the above example of $\mathrm{tr}(A^k)$ correspond to the power-sum polynomials $p_k$. But there are also the elementary symmetric polynomials $e_k$, which correspond to the characters of the exterior powers $\wedge^k V$, and the complete homogeneous symmetric polynomials $h_k$, which correspond to the characters of the symmetric powers $S^k V$, and of course the Schur polynomials $s_\lambda$ which are the characters of the irreducible representations of $\mathrm{GL}_n$.

***My Question:*** Are there analogs of Procesi's theorem, giving other generating sets of the ring of conjugation-invariant functions on $\mathrm{Mat}_n \times \cdots \times \mathrm{Mat}_n$, which specialize to the other classes of symmetric polynomials when $m=1$?