Unmanned aerial vehicles (UAVs) are capable of serving as flying base stations (BSs) for supporting data collection, artificial intelligence (AI) model training, and wireless communications. However, due to the privacy concerns of devices and limited computation or communication resource of UAVs, it is impractical to send raw data of devices to UAV servers for model training. Moreover, due to the dynamic channel condition and heterogeneous computing capacity of devices in UAV-enabled networks, the reliability and efficiency of data sharing require to be further improved. In this paper, we develop an asynchronous federated learning (AFL) framework for multi-UAV-enabled networks, which can provide asynchronous distributed computing by enabling model training locally without transmitting raw sensitive data to UAV servers. The device selection strategy is also introduced into the AFL framework to keep the low-quality devices from affecting the learning efficiency and accuracy. Moreover, we propose an asynchronous advantage actor-critic (A3C) based joint device selection, UAVs placement, and resource management algorithm to enhance the federated convergence speed and accuracy. Simulation results demonstrate that our proposed framework and algorithm achieve higher learning accuracy and faster federated execution time compared to other existing solutions.