  The Kibble-Zurek mechanism describes defect production due to non-adiabatic
passage through a critical point. Here we study its variant from ramping the
environment temperature to a critical point. We find that the defect density
scales as $\tau^{-d\nu}$ or $\tau^{-d/z}$ for thermal or quantum critical
points, respectively, in terms of the usual critical exponents and $1/\tau$ the
speed of the drive. Both scalings describe reduced defect density compared to
conventional Kibble-Zurek mechanism, which stems from the enhanced relaxation
due to bath-system interaction. Ramping to the quantum critical point is
investigated by studying the Lindblad equation for the transverse field Ising
chain in the presence of thermalizing bath, with couplings to environment
obeying detailed balance, confirming the predicted scaling. The von-Neumann or
the system-bath entanglement entropy follows the same scaling. Our results are
generalized to a large class of dissipative systems with power-law energy
dependent bath spectral densities as well.
