In this paper the kinetics of phase transitions of the first kind are studied in separable systems. Separable systems show short-range order but do not possess long-range order such as one-component, one- or two-dimensional solids or multicomponent solids of any dimension. The equilibrium and nonequilibrium phase transitions of the whole separable system have been characterized by the processes that take place within an independent subsystem of the separable system. The phase transition has been considered as a stochastic process where the order parameter of the transition is the stochastic variable. The nonlinear, coupled stochastic equations have been solved numerically by means of Adam's method. The processes were generated by three different types of temperature changes: linear heating/cooling, temperature jumps, and sinusoidal temperature oscillations. It was possible to carry out a comparative study of these processes within the framework of the same model. Processes from unstable to stable states start with increasing velocity, but after reaching an inflection point at time ti1, the processes slow down exponentially. When the initial unstable state is close to the equilibrium, ti1 is negligibly small compared to the time scale of the exponential decay and the whole process can be approximated by an exponential decay. This is the regime of the linear Onsager theory. Processes from metastable to stable states exhibit more complicated kinetics. The velocity of the process increases until the first inflection point at time ti1, and then the process slows down to a locally minimum velocity at ti2. Then the velocity increases again till the third inflection point at ti3 and finally slows down exponentially to zero velocity. Recently, these theoretically predicted kinetics were obtained experimentally, too, for the crystalline to gel phase transition of dipalmitoylphosphatidylcholine (DPPC) membranes. By means of the model, additional aspects of the phase transition kinetics have been studied, too, such as hysteresis, nucleation, homogeneous growth, relaxation processes, and phase and amplitude of steady-state structural oscillations. The model results are compared with the available experimental data taken from DPPC and dimiristoylphosphatidylcholine (DMPC) bilayer membranes.