Phase transformation via the nucleation and growth mechanism during a rapid cooling (quench) of a glass-forming liquid or melt is considered. Traditional approaches here are based on the quasi-steady-state (QSS) approximation for nucleation and the assumption of a size-independent growth. However, the QSS approach becomes invalid if the dimensionless rate of the barrier change, n = - tau partial derivative(W*/kT)/partial derivative t (tau is the inner time scale of the nucleation process) is not vanishingly small. For such "strongly time-dependent" situations an asymptotic (singular perturbation) technique of matching the time- and size-dependent nucleation and growth solutions is elaborated, and an explicit expression to describe the distribution function of large particles is derived. Formally, the results can be reproduced by the QSS approximation with the steady-state nucleation rate multiplied by an n-dependent factor. Analytical treatment is tested against numerically exact solutions of the nucleation (Becker-Doring) and growth equations.