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 $\ensuremath{\sigma}$ of current-voltage characteristics and transient current response to abrupt steps in applied voltage. For intermediate values of $\ensuremath{\sigma}$, 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 ${V}_{\mathit{\text{step}}}$; 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 $\ensuremath{\sigma}$ is identified above which the injected dipole mechanism is not observed for any value of ${V}_{\mathit{\text{step}}}$. Furthermore, at very low $\ensuremath{\sigma}$, we find a reversed static field configuration, i.e., with the high-field domain adjacent to the emitter contact.