We report on time-resolved photoluminescence studies of Forster resonance energy transfer (FRET) in structures with two arrays of epitaxial Cd(Zn)Se quantum dots (QDs) of different sizes separated by the ZnSe barrier of a variable width. The acceleration of recombination rate of both fast and slowly decaying components of emission from the energy-donating small QDs with the decrease of the barrier width is well consistent with the FRET mechanism. The found Forster radii turn out to be different for the fast and slow components. The rate acceleration is accompanied by the strong suppression of the slow emission component related, presumably, to the dark excitons. These findings open a way to control the characteristic of QD-based devices.