  A simple model of the driven motion of interacting particles in a two
dimensional random medium is analyzed, focusing on the critical behavior near
to the threshold that separates a static phase from a flowing phase with a
steady-state current. The critical behavior is found to be surprisingly robust,
being independent of whether the driving force is increased suddenly or
adiabatically. Just above threshold, the flow is concentrated on a sparse
network of channels, but the time scale for convergence to this fixed network
diverges with a larger exponent that that for convergence of the current
density to its steady-state value. This is argued to be caused by the
``dangerous irrelevance'' of dynamic particle collisions at the critical point.
Possible applications to vortex motion near to the critical current in dirty
thin film superconductors are discussed briefly.
