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