Static electric-field domains as well as current self-oscillations due to domain-wall oscillations have been observed in undoped, type-II GaAs-AlAs superlattices under photoexcitation. Photoluminescence measurements clearly demonstrate the coexistence of low- and high-field domains in the static and oscillating domain voltage regime. A comparison with the calculated energy level distribution indicates that the static domains are connected with negative differential velocity (NDV) due to resonant transfer between ${X}_{2}$ in the AlAs layer and ${\ensuremath{\Gamma}}_{1}$ in the GaAs layer, while the oscillating domains are attributed to NDV originating from resonant tunneling between ${X}_{1}$ and ${X}_{2}$ in the AlAs layers. This observation demonstrates the importance of transport channels due to the indirect band structure of type-II superlattices.