Currently, the registration of non-rigid 3D objects remains a challenging task. This paper proposes an unsupervised non-rigid point cloud registration network based on local correspondence relationships, namely the LCRNet. By learning the partial matching matrix to obtain more accurate point cloud correspondences and optimizing the matrix in combination with the Sinkhorn algorithm, we successfully eliminate the noise and outliers in the initial matching. In addition, through the adaptive learning of the transformer network and using complex geometric properties, we realize the deformation of non-rigid point clouds. The experiments on multiple datasets indicate that the LCRNet achieves accurate unsupervised non-rigid object registration, outperforming comparative methods.