Point cloud registration aligns multiple unposed point clouds into a common reference frame and is a core step for 3D reconstruction and robot localization without initial guess. In this work, we cast registration as conditional generation: a learned, continuous point-wise velocity field transports noisy points to a registered scene, from which the pose of each view is recovered. Unlike prior methods that perform correspondence matching to estimate pairwise transformations and then optimize a pose graph for multi-view registration, our model directly generates the registered point cloud, yielding both efficiency and point-level global consistency. By scaling the training data and conducting test-time rigidity enforcement, our approach achieves state-of-the-art results on existing pairwise registration benchmarks and on our proposed cross-domain multi-view registration benchmark. The superior zero-shot performance on this benchmark shows that our method generalizes across view counts, scene scales, and sensor modalities even with low overlap. Source code available at: https://github.com/PRBonn/RAP.