Robot-assisted dressing has the potential to significantly improve the lives of individuals with mobility impairments. To ensure an effective and comfortable dressing experience, the robot must be able to handle challenging deformable garments, apply appropriate forces, and adapt to limb movements throughout the dressing process. Prior work often makes simplifying assumptions -- such as static human limbs during dressing -- which limits real-world applicability. In this work, we develop a robot-assisted dressing system capable of handling partial observations with visual occlusions, as well as robustly adapting to arm motions during the dressing process. Given a policy trained in simulation with partial observations, we propose a method to fine-tune it in the real world using a small amount of data and multi-modal feedback from vision and force sensing, to further improve the policy's adaptability to arm motions and enhance safety. We evaluate our method in simulation with simplified articulated human meshes and in a real world human study with 12 participants across 264 dressing trials. Our policy successfully dresses two long-sleeve everyday garments onto the participants while being adaptive to various kinds of arm motions, and greatly outperforms prior baselines in terms of task completion and user feedback. Video are available at https://dressing-motion.github.io/.