  Using wall-crossing for K3 surfaces, we establish birational equivalence of
moduli spaces of stable objects on generic Enriques surfaces for different
stability conditions. As an application, we prove in the case of a Mukai vector
of odd rank that they are birational to Hilbert schemes. The argument makes use
of a new Chow-theoretic result, showing that moduli spaces on an Enriques
surface give rise to constant cycle subvarieties of the moduli spaces of the
covering K3.
