The aim of this M2 internship is to develop a quantum kinetic treatment of Bose gases quenched across their BEC transition, focusing on the non-perturbative regime where the gas's strongly correlated nature cannot be ignored. This scenario, recently realized experimentally, gives rise to a remarkable phenomenon of universal dynamic scaling, characterized by time-dependent scaling laws and dynamic exponents whose quantitative description remains to establish. This is the task that will be accomplished during this internship, using methods such as non-equilibrium quantum field theory and quantum kinetic equations. More generally, the internship will be an opportunity to delve into the modern research fields of non-equilibrium quantum physics and ultracold Bose gases.