We examine the depinning transitions and the temperature versus driving for
ce phase diagram for magnetically interacting pancake vortices in layered s
uperconductors. For strong disorder the initial depinning is plastic follow
ed by a sharp hysteretic transition to a three-dimensional ordered state fo
r increasing driving force. Our results are in good agreement with theoreti
cal predictions for driven anisotropic charge-density wave systems. We also
show that a temperature induced peak effect in the critical current occurs
due to the onset of plasticity between the layers.