We investigate continuity constraints on the plastic distortion and distortion rate tensors across surfaces of discontinuity moving through elastoplastic bodies. Tangential continuity of these tensors across interfaces moving normally to themselves derive from the conservation of the Burgers vector over patches bridging the interface, in the limit where such patches shrink onto the interface. Along interfaces whose motion involves a rotation, Burgers vector conservation follows from a trade-off between any tangential discontinuity of the dislocation-mediated plastic distortion rate relative to the interface and a plastic distortion rate arising from the rotation of the interface. The interface rotation dynamics follows from this balance. Additionally using the thermodynamic requirement of positive interface dissipation allows formulating mobility laws for the motion of the interface. The simplest admissible constitutive relationship relates the interface velocity to the traction vector. This mobility relationship recovers and develops in a tensorial context the conventional relations for grain boundary migration and grain growth in recovery/dynamic recrystallization processes, but challenges the constitutive character of the mobility parameter.