The kinetics of material diffusion across an interface is analyzed with special emphasis on the effect of the interface permeability on the transport characteristics, such as the mean-square displacement and the residence probability in each of the two phases separated by the interface. Two practically important cases are considered, namely, one-dimensional diffusion in a stepwise potential and three-dimensional diffusion under a spherically symmetric square-well potential. The transport characteristics are shown to be independent of the permeability at asymptotically long times, unless the interface is absolutely impermeable. However, the way the asymptotic behavior is approached can be controlled by the interface permeability. At low permeability values, the kinetics exhibit a characteristic intermediate stage that can last for such a long time that the true asymptotic behavior may not be observed experimentally at all. Along with exact analytical solutions of the diffusion problems at hand, useful approximate expressions are presented for different stages of the kinetics.