A theoretical approach for describing the kinetics of consecutive phase transformations ruled by nucleation and growth is reported. In the considered system, the mother phase (M) transforms to an intermediate phase (alpha) which, in turn, transforms to the final product (beta). The classical Kolmogorov-Johnson-Mehl-Avrami theory is generalized to deal with a finite-size phase with moving boundary. To this end, the statistical method based on the differential critical region has been employed. The exact solution of the kinetics is computed in closed form for the transformation of a spherical alpha-nucleus growing into the mother phase. By resorting to an approximate expression for the probability function entering the differential critical region method, the consecutive transformation is studied in the case of nucleation and growth of the alpha-phase. The time dependence of the beta/alpha volume fraction is found to be in very good agreement with the stretched exponential kinetics, and the dependence of Avrami's exponent on both nucleation and growth rates of the two phases is investigated. Modeling of the non-isothermal kinetics at constant heating rate has also been performed which provides an insight into the shape of the differential scanning calorimetry curves for consecutive phase transitions.