Recent technological breakthroughs have allowed unmanned aerial vehicles (UAVs) to be utilized in a broad range of new operations. Among these various applications, herein, we focus on the use of UAVs for search and rescue missions in emergency and postdisaster scenarios. In this context, self-charging technologies for drones create new challenges in the routing of UAVs with charging stations. We present a variant of the vehicle routing problem (VRP) to address the integrated use of UAVs and mobile charging stations and define the VRP with synchronized networks (VRPSN), a new class of VRPs involving the routing of UAVs whose recharge platforms can travel to different locations during an operation. This leads to two networks within the VRP that must be integrated and synchronized. This research develops a mixed-integer linear program model for the VRPSN that considers the use of UAVs and mobile charging stations in a synchronized manner. To overcome the computational limits of the MILP model, this research presents a construct-and-adjust heuristic method integrated with a genetic algorithm. As a numerical example, we test the proposed model on the Córrego do Feijão Mine located in Minas Gerais, Brazil, where a dam recently collapsed, killing many workers. Numerical tests show that the new methodology is an attractive planning method for providing efficient and rapid responses in search and rescue missions.