Although federated learning has achieved many breakthroughs recently, the heterogeneous nature of the learning environment greatly limits its performance and hinders its real-world applications. The heterogeneous data, time-varying wireless conditions and computing-limited devices are three main challenges, which often result in an unstable training process and degraded accuracy. Herein, we propose strategies to address these challenges. Targeting the heterogeneous data distribution, we propose a novel adaptive mixing aggregation (AMA) scheme that mixes the model updates from previous rounds with current rounds to avoid large model shifts and thus, maintain training stability. We further propose a novel staleness-based weighting scheme for the asynchronous model updates caused by the dynamic wireless environment. Lastly, we propose a novel CPU-friendly computation-reduction scheme based on transfer learning by sharing the feature extractor (FES) and letting the computing-limited devices update only the classifier. The simulation results show that the proposed framework outperforms existing state-of-the-art solutions and increases the test accuracy, and training stability by up to 2.38%, 93.10% respectively. Additionally, the proposed framework can tolerate communication delay of up to 15 rounds under a moderate delay environment without significant accuracy degradation.