  Domain wall dynamics in ferromagnets is complicated by internal degrees of
freedom of the domain walls. We develop a model of domain walls in disordered
thin films with perpendicular magnetic anisotropy capturing such features, and
use it to study the depinning transition. For weak disorder, excitations of the
internal magnetization are rare, and the depinning transition takes on exponent
values of the quenched Edwards-Wilkinson equation. Stronger disorder results in
disorder-dependent exponents concurrently with nucleation of an increasing
density of Bloch lines within the domain wall.
