Unmanned aerial vehicles (UAVs) can enhance the performance of cellular networks, due to their high mobility and efficient deployment. In this paper, we present a first study on how the user mobility affects the UAVs' trajectories of a multiple-UAV assisted wireless communication system. Specifically, we consider the UAVs are deployed as aerial base stations to serve ground users who move between different regions. We maximize the throughput of ground users in the downlink communication by optimizing the UAVs' trajectories, while taking into account the impact of the user mobility, propulsion energy consumption, and UAVs' mutual interference. We formulate the problem as a route selection problem in an acyclic directed graph. Each vertex represents a task associated with a reward on the average user throughput in a region-time point, while each edge is associated with a cost on the energy propulsion consumption during flying and hovering. For the centralized trajectory design, we first propose the shortest path scheme that determines the optimal trajectory for the single UAV case. We also propose the centralized route selection (CRS) scheme to systematically compute the optimal trajectories for the more general multiple-UAV case. Due to the NP-hardness of the centralized problem, we consider the distributed trajectory design that each UAV selects its trajectory autonomously and propose the distributed route selection (DRS) scheme, which will converge to a pure strategy Nash equilibrium within a finite number of iterations.