Numerical simulation results are presented for a discrete drift-diffusion rate equation model that describes electronic transport due to sequential tunneling between adjacent quantum wells in weakly coupled semiconductor superlattices. We study the dependence on contact conductivity σ of current-voltage characteristics and transient current response to abrupt steps in applied voltage. For intermediate values of σ, three qualitatively distinct transient responses—each associated with a different mechanism for the relocation of a static charge accumulation layer—are observed for different values of voltage step Vstep; these involve, respectively, (1) the motion of a single charge accumulation layer, (2) the motion of an injected charge dipole, and (3) the motion of an injected monopole. A critical value of σ is identified above which the injected dipole mechanism is not observed for any value of Vstep. Furthermore, at very low σ, we find a reversed static field configuration, i.e., with the high-field domain adjacent to the emitter contact.