Existing learning-based Vehicle Routing Problem (VRP) solvers typically assume static edge costs, ignoring the temporal variability of road traffic. In ambulance dispatch, however, travel times fluctuate markedly with rush-hour congestion and incidents, so routes that look optimal offline can delay patient delivery. We introduce TraffiX-MoE, a neural solver that couples a traffic-aware simulator with a Mixture-of-Experts (MoE) Transformer. TraffiX-MoE represents time-varying edge costs via slot-indexed tensors, augments POMO with expert specialisation for distinct congestion regimes, and trains with a latency-controlled hierarchical gating scheme. Experiments on synthetic and real Adelaide traffic show that TraffiX-MoE cuts average evacuation time by 8–11% over strong baselines while retaining sub-second planning latency.