  Under some mild assumptions, an orientation-preserving branched covering map
of marked $2$-spheres induces a pullback map between the corresponding
Teichm\"uller spaces. By analyzing the associated pushforward operator acting
on integrable quadratic differentials, we obtain a global lower bound on the
rank of the derivative of the pullback map in terms of the action of the cover
on the marked points. In the dynamical context, the two sets of marked points
in the target and source coincide with the postcritical set. Investigating the
resulting pullback map is the central part of Thurston's topological
characterization of postcritically finite rational maps. Postcritically finite
maps with constant pullback have been studied by various authors. In that
direction, our approach provides upper bounds on the size of the postcritical
set of a map with constant pullback, and shows that the postcritical dynamics
is highly restricted.
