We consider current-induced domain wall motion and, the reciprocal process, moving domain wall-induced current. The associated Onsager coefficients are expressed in terms of scattering matrices. Uncommonly, in (Ga,Mn)As, the effective Gilbert damping coefficient $α_w$ and the effective out-of-plane spin transfer torque parameter $β_w$ are dominated by spin-orbit interaction in combination with scattering off the domain wall, and not scattering off extrinsic impurities. Numerical calculations give $α_w ∼ 0.01$ and $β_w ∼ 1$ in dirty (Ga,Mn)As. The extraordinary large $β_w$ parameter allows experimental detection of current or voltage induced by domain wall motion in (Ga,Mn)As. ; Comment: Final version accepted by Physical Review Letters