In this paper, the maximal dynamic expected flows (MDEF) problem, which seeks the paths and associated flows that maximize the expected number of evacuees who successfully egress within a given time bound T, is defined. The MDEF problem is considered in a network, where arc traversal times and capacities are discrete random variables with time-varying distribution functions and capacities are assumed to be recaptured over time (i.e., the network is dynamic). A metaheuristic based on the principles of noisy genetic algorithms is proposed for its solution. Details of a novel approach (using an approximate likelihood measure) for overcoming the difficulties associated with computing state probabilities that require extremely small floating-point arithmetic are provided. Results from numerical experiments run on randomly generated networks with as many as 500 nodes are given. The results indicate that the proposed solution technique and approximate likelihood measure perform well. Solution of the proposed problem formulation results in robust evacuation paths that are likely to save the largest number of people before conditions become untenable.