A mathematical crystallization model in the meso scale (the intermediate dimension scale between interatomic distance in solids and grain size in metals and alloys) is presented with the use of a kinetic-diffusion cellular automaton model. The model considers the non-equilibrium character of real processes of phase transformations, where the kinetic undercooling of the solid-liquid interface is a measure of this non-equilibrium level. Anisotropy of the interface mobility is assumed. The modelling results are compared to the experimental data.