We study the spin injection dynamics and the effects of spin relaxation in self-assembled quantum dots (QDs) of CdSe coupled with a diluted magnetic semiconductor (DMS) layer of ZnMnSe, where spin-polarized excitons can be injected from the DMS into the QDs because of the potential difference. The degree of circular polarization, $P$, of excitonic photoluminescence (PL) at 5 T in the coupled QDs shows a rapid increase with increasing delay time after a linearly polarized pulse excitation, indicating the spin-injection dynamics. The $P$ value tends to decay gradually because of the exciton-spin relaxation in the QDs after the spin injection. The spin-polarized excitons in the QD ensemble migrate simultaneously from QDs with higher exciton energies to those with lower exciton energies. This inter-dot transfer of excitons also affects the $P$ value in the lower energy region of the QD-emission band because the excitons lose their spin polarizations due to the spin relaxation in the dots during the migration. The detailed analysis for the exciton-spin transfer is presented in a coupled QD system after spin injection.