We show that the Dexter-Förster-like radiationless resonance energy-transfer process from excitonic states into the Mn 3d 5 shell in wide-gap ͑II,Mn͒VI semiconductors is strongly enhanced if the total spin of the overall process is conserved. This requirement cannot be fulfilled in processes involving a bright exciton, i.e., other excitonic complexes or processes need to be involved. Of these we discuss dark excitons, donor-bound excitons D 0 X, negatively charged excitons X Ϫ , and an Auger-like process. A careful analysis of the magnetic field and time dependence of the excitonic and Mn luminescence in the ͑Zn,Cd,Mn͒Se samples under study gives evidence that the D 0 X complex plays a dominant role in the intralayer energy transfer. In asymmetric double quantum well ͑ADQW͒ structures consisting of a ͑Zn,Cd͒Se well and a ͑Zn,Cd,Mn͒Se well embedded in ZnSe barriers, there is a competition between the intralayer and interlayer energy-transfer processes from excitonic states into the Mn system. These interlayer processes take place between the Mn ions situated in the ͑Zn,Cd,Mn͒Se well and spatially indirect excitons ͓where the hole is confined in the ͑Zn,Cd,Mn͒Se and the electron confined in the ͑Zn,Cd͒Se͔ as well as spatially direct excitons of the ͑Zn,Cd͒Se well. The continuous magnetic-field tuning demonstrates convincingly the subtle interplay of excitonic band structure of the ADQW and spin effects in the energy-transfer processes. A spin-dependent energy transfer should be a general feature of rare-earth or transition-metal doped semiconductors such as II-VI or III-V semiconductors or even Er-doped Si or SiO 2 .