A complete analytical solution of the integro-differential model describing the transient nucleation and growth of the crystals at the intermediate stage of phase transitions is constructed. The roles of external heat/mass sources appearing in the balance equations and the crystallite withdrawal rate entering in the Fokker-Planck equation are detailed. An exact analytical solution of the Folder-Planck equation is found for arbitrary nucleation mechanisms and growth kinetics. Two important cases of the Weber-Volmer-Frenkel-Zaclovich and Meirs kinetics are considered in some detail. A non-linear time dependent integral equation with memory kernel for the metastability level is analytically solved on the basis of the saddle point method for the Laplace integral in the case of mixed kinetic diffusion regime of crystal growth, which is of frequent occurrence. It is shown that the desupercooling/desupersaturation rate decreases with increasing the crystal withdrawal rate and intensities of external sources. The density distribution function becomes more and more broad with time. In addition, this function increases with decreasing the crystallite withdrawal rate and with increasing intensities of external sources. (C) 2014 Elsevier Ltd. All rights reserved.