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Let $\Gamma$ and $\Delta$ be sufficiently distinct countable groups. We showthat there is an orbit equivalence relation $E$, induced by an action of thePolish 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 Clemensand Coskey. The proofs follow a translation between Borel homomorphisms anddefinable pins.

Strong ergodicity for the Gamma-jump operator and for actions of Polish wreath products