For a moduli space $M$ of stable sheaves over a K3 surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\ast (M\times X^l)$, $l\geq 1$, generalizing the classic Beauville-Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\ast (M)\subset CH_\ast (M)$. We prove the proposed identities when $M$ is the Hilbert scheme of points on a K3 surface. This is joint work with L. Flapan, A. Marian and R. Silversmith.