  Constant cycle curves on a K3 surface $X$ over $\mathbb{C}$ have been
introduced by Huybrechts (2014) as curves whose points all define the same
class in the Chow group. In this paper we study correspondences $Z \subseteq
X\times X$ over $\mathbb{C}$ acting on the group $\mbox{ccc}(X)$ of cycles
generated by irreducible constant cycle curves. We construct for any $n\geq 2$
and any very ample line bundle $L$ a locus $Z_n(L)\subseteq X\times X$ of
expected dimension $2$, which yields a correspondence that acts on
$\mbox{ccc}(X)$, when it has the expected dimension. We provide examples of
$Z_n(L)$ for low $n$ and exhibit one correspondence different from $Z_n(L)$
acting on $\mbox{ccc}(X)$.
