We study the spin and charge dynamics of electrons in n-doped II--VI semiconductor multiple quantum wells when one or more quantum wells are doped with Mn. The interplay between strongly nonlinear inter-well charge transport and the large tunable spin-splitting induced by exchange interactions with spin-polarized Mn ions produces interesting new spin dependent features. The tunneling current between quantum wells can be strongly spin polarized and, under certain conditions, can develop self sustained oscillations under a finite dc voltage. The spin polarization oscillates in both magnetic and nonmagnetic quantum wells and the time average in magnetic wells can differ from its zero-voltage value. Our numerical simulations demonstrate that the amplitude of the spin polarization oscillations depends on the distribution of magnetic wells within the sample. We discuss how the spin polarized current and the spin polarization of the quantum wells can be tailored experimentally.