To exploit unprecedented data generation in mobile edge networks, federated learning (FL) has emerged as a promising alternative to the conventional centralized machine learning (ML). However, there are some critical challenges for FL deployment. One major challenge called straggler issue severely limits FL's coverage where the device with the weakest channel condition becomes the bottleneck of the model aggregation performance. Besides, the huge uplink communication overhead compromises the effectiveness of FL, which is particularly pronounced in large-scale systems. To address the straggler issue, we propose the integration of an unmanned aerial vehicle (UAV) as the parameter server (UAV-PS) to coordinate the FL implementation. We further employ over-the-air computation technique that leverages the superposition property of wireless channels for efficient uplink communication. Specifically, in this paper, we develop a novel UAV-enabled over-the-air asynchronous FL (UAV-AFL) framework which supports the UAV-PS in updating the model continuously to enhance the learning performance. Moreover, we conduct a convergence analysis to quantitatively capture the impact of model asynchrony, device selection and communication errors on the UAV-AFL learning performance. Based on this, a unified communication-learning problem is formulated to maximize asymptotical learning performance by optimizing the UAV-PS trajectory, device selection and over-the-air transceiver design. Simulation results demonstrate that the proposed scheme achieves substantially learning efficiency improvement compared with the state-of-the-art approaches.