Reliable matching between partially observed, deforming point clouds requires global context and fine geometric detail. Coarse candidate selection can exclude correct fine-level correspondences. We present \paper, a unified conditional transport framework with the matching matrix itself as the evolving state. Coarse diffusion establishes global matching hypotheses; hierarchy-preserving lifting expands them into a structured high-resolution source. Geometry-conditioned ODE and SDE bridges continue refinement in the complete fine-level candidate space, allowing coarse errors to be corrected. The deterministic endpoint-parameterized conditional flow matching (CFM) design improves matching through iteration, while the SDE forward drift enables effective few-step refinement. Experiments on 4DMatch and 4DLoMatch demonstrate competitive matching and non-rigid registration, with transfer to CAPE and DeepDeform without target-domain training.

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