  Let $\phi$ be a post-critically finite branched covering of a two-sphere. By
work of Koch, the Thurston pullback map induced by $\phi$ on Teichm\"{u}ller
space descends to a multi-valued self-map --- a Hurwitz correspondence
$\mathcal{H}_{\phi}$ --- of the moduli space $\mathcal{M}_{0,P}$. We study the
dynamics of Hurwitz correspondences via numerical invariants called dynamical
degrees. We show that the sequence of dynamical degrees of $\mathcal{H}_{\phi}$
is always non-increasing, and the behavior of this sequence is constrained by
the behavior of $\phi$ at and near points of its post-critical set.
