Reliable non-rigid point cloud correspondences are important for deformable anatomical registration, embodied perception and manipulation, and dynamic 3D reconstruction. Coarse-to-fine methods reduce computational cost by selecting the top-
K coarse regions. However, this pruning may remove weak but correct hypotheses and restrict fine matching to an incomplete search space. We present \paper, a two-stage generative solver that maintains the complete soft matching matrix at both coarse and high resolutions. Stage
I uses denoising diffusion to estimate a global matching matrix in the compact coarse-resolution space. We then lift this matrix to high resolution while preserving its hierarchy. The lifted matrix is rank-bounded and block-constant. StageII refines it through a conditional transport bridge. We implement the bridge with two types of dynamics: a deterministic endpoint-parameterized conditional Flow Matching (CFM) ODE and a stochastic Brownian-bridge SDE inspired by Schr"odinger bridges. Both variants share the lifted source, a time-conditioned transformer, and a matching-matrix endpoint predictor. Experiments on 4DMatch and 4DLoMatch show that both variants produce more accurate correspondences than the compared methods and improve downstream registration, with larger gains in low-overlap cases. They also improve cross-dataset generalization on CAPE and DeepDeform without target-domain adaptation while using the same deformation solver.