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The response of an n-doped dc voltage biased II-VI multi-quantum well dilute magnetic semiconductor nanostructure having its first well doped with magnetic (Mn) impurities is analyzed by sweeping wide ranges of both the voltage and the Zeeman level splitting induced by an external magnetic field. The level splitting versus voltage phase diagram shows regions of stable self-sustained current oscillations immersed in a region of stable stationary states. Transitions between stationary states and self-sustained current oscillations are systematically analyzed by both voltage and level splitting abrupt switching. Sudden voltage or/and magnetic field changes may switch on current oscillations from an initial stationary state, and reciprocally, current oscillations may disappear after sudden changes of voltage or/and magnetic field changes into the stable stationary states region. The results show how to design such a device to operate as a spin injector and a spin oscillator by tuning the Zeeman splitting (through the applied external magnetic field), the applied voltage and the sample configuration parameters (doping density, barrier and well widths, etc.) to select the desired stationary or oscillatory behavior.