  Let $\phi:S^2 \to S^2$ be an orientation-preserving branched covering whose
post-critical set has finite cardinality $n$. If $\phi$ has a fully ramified
periodic point $p_{\infty}$ and satisfies certain additional conditions, then,
by work of Koch, $\phi$ induces a meromorphic self-map $R_{\phi}$ on the moduli
space $\mathcal{M}_{0,n}$; $R_{\phi}$ descends from Thurston's pullback map on
Teichm\"uller space. Here, we relate the dynamics of $R_{\phi}$ on
$\mathcal{M}_{0,n}$ to the dynamics of $\phi$ on $S^2$. Let $\ell$ be the
length of the periodic cycle in which the fully ramified point $p_{\infty}$
lies; we show that $R_{\phi}$ is algebraically stable on the heavy-light
Hassett space corresponding to $\ell$ heavy marked points and $(n-\ell)$ light
points.
