Kinetics of supersaturated solutions with restricted size of precipitating particles is considered using the microscopic theory which includes monomer-monomer agglomeration (MMA) and dimer adsorption/emission (DAE). It is shown that the system passes, during its evolution from initial to final state, through four consecutive stages. In stage I and stage III, the scaled size distribution of particles changes quickly with increasing time while in stage II and stage IV it is almost independent of time. It is argued that during stage II the system, at any rates of MMA and DAE, is in a regime similar to the asymptotic regime provided by the classical Lifshitz-Slyouzov-Wagner (LSW) theory where no restrictions on the size of the particles are imposed and MMA and DAE are not considered. Extrapolating the time dependence of mean radius (R) over bar of particles, obtained in range II, to large times provides the power law (R) over bar similar to t(1/v) where v depends on the rates of MMA and DAE and changes from v = 2 when MMA and DAE are absent and v = 2.3 for comparatively large rates of MMA and DAE. This complements the result of the classical LSW approach where change of v was attributed to the relationship between the rate of particles' diffusion and rate of adsorption of monomers on the surface of larger particles. During stage IV (large times), the system approaches exponentially the asymptotic state which is different from that for classical theory. The time behavior of the concentrations of the particles of largest size and of mean radius are examined for both asymptotic regimes.