A theoretical model is constructed that describes diffusion processes enhanced by the elastic interaction of the diffusing species and amorphous grain boundaries in nanocrystalline and polycrystalline solids. A partial differential equation governing evolution of the spatially inhomogeneous concentration of the diffusing species near amorphous grain boundaries is numerically solved. It is shown that the enhanced diffusion essentially influences the growth of amorphous grain boundaries in solids under irradiation and thermal treatment as well in sintered ceramics and is capable of contributing to the anomalously fast diffusion in nanocrystalline solids.