The crystallization of a solid generally results in the formation of a microstructure of grains with various sizes and shapes. A theory has been introduced recently to describe this inhomogeneous state through the derivation of the non-equilibrium grain size distribution from early stages to full crystallization. The theory assumed an isotropic growth rate and the formation of nuclei with specific size. We discuss two generalizations of the theory. In the first generalization, we replace the isotropic growth rate by an anisotropic rate that leads to the formation of ellipsoidal grains before impingement. We obtain an analytical expression for the grain size distribution for the case where the growth rate leads to a change in volume leaving the shape of unimpinged grains invariant. We present the case for a d-dimensional solid with d = 1 (wire), d = 2 (thin films), and d = 3 (bulk solid). In the second generalization, we study how the distribution for isotropic growth is affected by replacing the Dirac delta-type source term of nuclei by a more physical Gaussian-type source. Our results show that in both generalizations the grain size distribution remains essentially unaffected. The two generalizations provide similar results as the isotropic case.