Evacuations necessitated by extreme events are usually envisioned as taking place with all people evacuating simultaneously, thereby leading to premature congestion on the surface streets and excessive delays. By staggering the evacuating load onto the network, the onset of congestion may be delayed, and people evacuate faster. In this paper, we consider the problem of scheduling evacuation trips between a selected set of origin nodes and (safety) destinations, with the objective of minimizing the network clearance time for the evacuees while minimizing the disruption to the network. We propose an iterative bi-level formulation framework for solving this problem, whereby a dynamic network assignment problem is solved in the upper level to determine the time-dependent route assignments (shipments) and a dynamic loading problem is solved in the lower level to determine the corresponding route travel times. The method of successive averages is used to solve the upper level, while DYNASMART-P, a simulation-based dynamic traffic assignment model, is used to solve the lower problem. The model determines the departure time, route, and destination for each evacuee, which is aggregated to produce a time-dependent staging policy for each origin.