  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 {\it 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 Z_2 topologically ordered phase of the toric code/Z_2 gauge
theory, and the non-abelian SU(2)$_k$ ordered phases of the relevant Levin-Wen
models.
