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Hurwitz correspondences are certain multivalued self-maps of the moduli space$\mathcal{M}_{0,N}$. They arise in the study of Thurston's topologicalcharacterization of rational functions. We consider the dynamics of Hurwitzcorrespondences 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 aweighted stable curves compactification $X_N^\dagger$ that is optimal for itsdynamics. 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 thepushforward induced by $\mathcal{H}$ on a natural quotient of$H_{2k}(\overline{\mathcal{M}}_{0,N})$.

Hurwitz correspondences on compactifications of $\mathcal{M}_{0,N}$