This paper introduces a 3D point cloud sequence learning model based on inconsistent spatio-temporal propagation for LiDAR odometry, termed DSLO. It consists of a pyramid structure with a spatial information reuse strategy, a sequential pose initialization module, a gated hierarchical pose refinement module, and a temporal feature propagation module. First, spatial features are encoded using a point feature pyramid, with features reused in successive pose estimations to reduce computational overhead. Second, a sequential pose initialization method is introduced, leveraging the high-frequency sampling characteristic of LiDAR to initialize the LiDAR pose. Then, a gated hierarchical pose refinement mechanism refines poses from coarse to fine by selectively retaining or discarding motion information from different layers based on gate estimations. Finally, temporal feature propagation is proposed to incorporate the historical motion information from point cloud sequences, and address the spatial inconsistency issue when transmitting motion information embedded in point clouds between frames. Experimental results on the KITTI odometry dataset and Argoverse dataset demonstrate that DSLO outperforms state-of-the-art methods, achieving at least a 15.67\% improvement on RTE and a 12.64\% improvement on RRE, while also achieving a 34.69\% reduction in runtime compared to baseline methods. Our implementation will be available at https://github.com/IRMVLab/DSLO.