  Hurwitz correspondences are certain multivalued self-maps of the moduli space
$\mathcal{M}_{0,N}$. They arise in the study of Thurston's topological
characterization of rational functions. We consider the dynamics of Hurwitz
correspondences and ask: On which compactifications of $\mathcal{M}_{0,N}$
should they be studied? We compare a Hurwitz correspondence $\mathcal{H}$
across various modular compactifications of $\mathcal{M}_{0,N}$, and find a
weighted stable curves compactification $X_N^\dagger$ that is optimal for its
dynamics. We use $X_N^\dagger$ to show that the $k$th dynamical degree of
$\mathcal{H}$ is the absolute value of the dominant eigenvalue of the
pushforward induced by $\mathcal{H}$ on a natural quotient of
$H_{2k}(\overline{\mathcal{M}}_{0,N})$.
