  We use wall-crossing in the Bridgeland stability manifold to systematically
study the birational geometry of the moduli space $M_\sigma(\mathbf{v})$ of
$\sigma$-semistable objects of class $\mathbf{v}$ for a generic stability
condition $\sigma$ on an arbitrary Enriques surface $X$. In particular, we show
that for any other generic stability condition $\tau$, the two moduli spaces
$M_\tau(\mathbf{v})$ and $M_\sigma(\mathbf{v})$ are birational. As a
consequence, we show that for primitive $\mathbf{v}$ of odd rank
$M_\sigma(\mathbf{v})$ is birational to a Hilbert scheme of points. Similarly,
in even rank we show that $M_\sigma(\mathbf{v})$ is birational to a moduli
space of torsion sheaves supported on a hyperelliptic curve when
$\ell(\mathbf{v})=1$. As an added bonus of our work, we prove that the
Donaldson-Mukai map
$\theta_{\mathbf{v},\sigma}:\mathbf{v}^\perp\to\mathrm{Pic}(M_\sigma(\mathbf{v}))$
is an isomorphism for these classes. Finally, we use our classification to
fully describe the geometry of the only two examples of moduli of stable
sheaves on $X$ that are uniruled (and thus not K-trivial).
