The interplay of tunneling transport and carrier-mediated ferromagnetism in narrow semiconductor multiquantum well structures containing layers of GaMnAs is investigated within a self-consistent Green's function approach, accounting for disorder in the Mn-doped regions and unwanted spin flips at heterointerfaces on phenomenological ground. We find that the magnetization in GaMnAs layers can be controlled by an external electric bias. The underlying mechanism is identified as spin-selective hole tunneling in and out of the Mn-doped quantum wells, whereby the applied bias determines both hole population and spin polarization in these layers. In particular, we predict that, near resonance, ferromagnetic order in the Mn-doped quantum wells is destroyed. The interplay of both magnetic and transport properties combined with structural design potentially leads to several interrelated physical phenomena, such as dynamic spin filtering, tunneling-induced bias anomaly, electrical control of magnetization in individual magnetic layers, and, under specific bias conditions, to self-sustained current and magnetization oscillations (magnetic multistability). Relevance to recent experimental results is discussed.