  Let $\Gamma$ and $\Delta$ be sufficiently distinct countable groups. We show
that there is an orbit equivalence relation $E$, induced by an action of the
Polish wreath product group $\Gamma\wr\Gamma$, so that $E$ is generically
$F$-ergodic for any orbit equivalence relation $F$ induced by an action of
$\Delta\wr\Delta$. More generally, we establish generic ergodicity between
$\Gamma$-jumps and the iterated $\Delta$-jumps, answering a question of Clemens
and Coskey. The proofs follow a translation between Borel homomorphisms and
definable pins.
