  We study the higher ramification structure of dynamical branch extensions,
and propose a connection between the natural dynamical filtration and the
filtration arising from the higher ramification groups: each member of the
former should, after a linear change of index, coincide with a member of the
latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie
extensions. By explicitly calculating the Hasse-Herbrand functions of such
branch extensions, we are able to show that this description is accurate for
some families of polynomials, in particular post-critically bounded polynomials
of $p$-power degree. We apply our results to give a partial answer to a
question of Berger (in arXiv:1411.7064) and a partial answer to a question
about wild ramification in arboreal extensions of number fields (raised in both
arXiv:math/0408170 and arXiv:1511.00194).
