To assist the design of efficient molecular junctions, a precise understanding of the charge transport mechanisms through nanoscaled devices is of prime importance. In the present contribution, we present time- and space-resolved electron transport simulations through a nanojunction under time-dependent potential biases. We use the driven Liouville-von Neumann approach to simulate the time evolution of the one-electron density matrix under nonequilibrium conditions, which allows us to capture the ultrafast scattering dynamics, the electronic relaxation process, and the quasi-stationary current limit from the same simulation. Using local projection techniques, we map the coherent electronic current density, unraveling insightful mechanistic details of the transport on time scales ranging from atto- to picoseconds. Memory effects dominate the early time transport process, and they reveal different current patterns on short time scales in comparison to those in the long-time regime. For nanotransistors with high switching rates, the scattering perspective on electron transport should thus be favored.