The emergence of a finite-size ordered equilibrium quantum state from an uncorrelated thermal state through the dynamical crossing of a phase transition is a central question of non-equilibrium many-body physics. During such crossing, the system breaks its symmetry by establishing numerous uncorrelated regions separated by spontaneously-generated defects (e.g. solitons, or vortices), whose emergence obeys a universal Kibble-Zurek scaling law with the quench duration. Much less is known about the ensuing re-equilibrating or “coarse-graining" stage, which is governed by the evolution and interactions of such defects under system-specific and external constraints. Motivated by recent experiments in a temperature-quenched elongated ultracold atomic gas, we perform a detailed numerical study of condensate growth and re-equilibration dynamics. Our work provides a unique numerical visualization of the entire non-equilibrium process, demonstrating direct agreement with experimental findings and the quench-induced decoupling of number and coherence growth.