We consider the non-equilibrium dynamics of topologically ordered systems driven across a continuous phase transition into proximate phases with no, or reduced, topological order. This dynamics exhibits scaling in the spirit of Kibble and Zurek but now without the presence of symmetry breaking and a local order parameter. The late stages of the process are seen to exhibit a slow, coarsening dynamics for the string-net that underlies the physics of the topological phase, a potentially interesting signature of topological order. We illustrate these phenomena in the context of particular phase transitions out of the Abelian Z2 topologically ordered phase of the toric code/Z2 gauge theory, and the non-Abelian SU(2)k ordered phases of the relevant Levin-Wen models.