Taking into account the coupling between the position of the wall and an internal degree of freedom, namely, its phase φ, we examine, in the rigid-wall approximation, the dynamics of a magnetic domain wall subject to a weak pinning potential. We determine the corresponding force-velocity characteristics, which display several unusual features when compared to the standard depinning behavior. At zero temperature, there exists a bistable regime for low forces, with a logarithmic behavior close to the transition. For weak pinning, there occurs a succession of bistable transitions corresponding to different topological modes of the phase evolution. At finite temperature, the force-velocity characteristics become nonmonotonous. We compare our results to recent experiments on permalloy nanowires.