In the framework of the theory of phase transformations with position-dependent nucleation rate, a model has been developed aimed at describing the dissolution-precipitation reaction at the surface of small particles. The precipitation reaction takes place by nucleation and growth processes under time-dependent supersaturation. Depending on the coverage of the particle surface by the new phase, the reaction kinetics exhibits high- and low-rate regimes. The computation is performed for both progressive and simultaneous nucleation. In the case of simultaneous nucleation, closed-form solutions are attained for diffusion- and interface-limited growth modes and for isotropic and anisotropic growths of the nuclei, as well. The scaling properties of the kinetics on particle size are also investigated. The kinetic model is employed for analysing experimental data and makes it possible to estimate the nucleation density on the particle surface and to have an insight into the microscopic growth law of nuclei.