Dexterous grasping of unseen objects in dynamic environments is an essential prerequisite for the advanced manipulation of autonomous robots. Prior advances rely on several assumptions that simplify the setup, including environment stationarity, pre-defined objects, and low-dimensional end-effectors. Though easing the problem and enabling progress, it undermined the complexity of the real world. Aiming to relax these assumptions, we present a dynamic grasping framework for unknown objects in this work, which uses a five-fingered hand with visual servo control and can compensate for external disturbances. To establish such a system on real hardware, we leverage the recent advances in real-time dexterous generative grasp synthesis and introduce several techniques to secure the robustness and performance of the overall system. Our experiments on real hardware verify the ability of the proposed system to reliably grasp unknown dynamic objects in two realistic scenarios: objects on a conveyor belt and human-robot handover. Note that there has been no prior work that can achieve dynamic multi-fingered grasping for unknown objects like ours up to the time of writing this paper. We hope our pioneering work in this direction can provide inspiration to the community and pave the way for further algorithmic and engineering advances on this challenging task. A video of the experiments is available at https://youtu.be/b87zGNoKELg.