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As a promising distributed machine learning paradigm that enables collaborative training without compromising data privacy, Federated Learning (FL) has been increasingly used in AIoT (Artificial Intelligence of Things) design. However, due to the lack of efficient management of straggling devices, existing FL methods greatly suffer from the problems of low inference accuracy and long training time. Things become even worse when taking various uncertain factors (e.g., network delays, performance variances caused by process variation) existing in AIoT scenarios into account. To address this issue, this paper proposes a novel asynchronous FL framework named GitFL, whose implementation is inspired by the famous version control system Git. Unlike traditional FL, the cloud server of GitFL maintains a master model (i.e., the global model) together with a set of branch models indicating the trained local models committed by selected devices, where the master model is updated based on both all the pushed branch models and their version information, and only the branch models after the pull operation are dispatched to devices. By using our proposed Reinforcement Learning (RL)-based device selection mechanism, a pulled branch model with an older version will be more likely to be dispatched to a faster and less frequently selected device for the next round of local training. In this way, GitFL enables both effective control of model staleness and adaptive load balance of versioned models among straggling devices, thus avoiding the performance deterioration. Comprehensive experimental results on well-known models and datasets show that, compared with state-of-the-art asynchronous FL methods, GitFL can achieve up to 2.64X training acceleration and 7.88% inference accuracy improvements in various uncertain scenarios.