Unmanned Aerial Vehicle (UAV) networks are essential for data transmission in emergency scenarios, serving as relays to transmit data from ground users to base stations. Traditional UAV routing focuses primarily on unicast routing, which requires that a specific destination be identified for the data before transmission begins. This approach encounters significant challenges in dynamic networks due to frequent topology changes. When multiple base stations are available within the network, routing data to several base stations can enhance transmission efficiency. However, existing routing algorithms are not well-suited for such scenarios. This paper redefines the routing of UAV networks with multiple base stations as anycast routing tailored for dynamic networks. We introduce a distributed anycast routing method named QAR to boost data transmission performance. In the QAR, Q-learning parameters are dynamically adjusted, and a multi-base station transmission value function is crafted to calculate rewards and update the Q-table. Simulation results indicate that QAR surpasses existing Q-learning based routing methods in multiple base station scenarios, delivering superior performance in terms of delay, packet delivery ratio, and throughput.