Navigating unmanned aerial vehicles (UAVs) through cluttered and dynamic environments remains a significant challenge, particularly when dealing with fast-moving or sudden-appearing obstacles. This paper introduces a complete LiDAR-based system designed to enable UAVs to avoid various moving obstacles in complex environments. Benefiting the high computational efficiency of perception and planning, the system can operate in real time using onboard computing resources with low latency. For dynamic environment perception, we have integrated our previous work, M-detector, into the system. M-detector ensures that moving objects of different sizes, colors, and types are reliably detected. For dynamic environment planning, we incorporate dynamic object predictions into the integrated planning and control (IPC) framework, namely DynIPC. This integration allows the UAV to utilize predictions about dynamic obstacles to effectively evade them. We validate our proposed system through both simulations and real-world experiments. In simulation tests, our system outperforms state-of-the-art baselines across several metrics, including success rate, time consumption, average flight time, and maximum velocity. In real-world trials, our system successfully navigates through forests, avoiding moving obstacles along its path.