  We study the dynamic after a smooth quench across a continuous transition
from the disordered phase to the ordered phase. Based on scaling ideas, linear
response and the spectrum of unstable modes, we develop a theoretical
framework, valid for any second order phase transition, for the early-time
evolution of the condensate in the broken phase. Our analysis unveils a novel
period of non-adiabatic evolution after the system passes through the phase
transition, where a parametrically large amount of coarsening occurs before a
well-defined condensate forms. Our formalism predicts a rate of defect
formation parametrically smaller than the Kibble-Zurek prediction and yields a
criterion for the break-down of Kibble-Zurek scaling for sufficiently fast
quenches. We numerically test our formalism for a thermal quench in a 2 + 1
dimensional holographic superfluid. These findings, of direct relevance in a
broad range of fields including cold atom, condensed matter, statistical
mechanism and cosmology, are an important step towards a more quantitative
understanding of dynamical phase transitions.
