Understanding motion in dynamic environments is critical for autonomous driving, thereby motivating research on class-agnostic motion prediction. In this work, we investigate weakly and self-supervised class-agnostic motion prediction from LiDAR point clouds. Outdoor scenes typically consist of mobile foregrounds and static backgrounds, allowing motion understanding to be associated with scene parsing. Based on this observation, we propose a novel weakly supervised paradigm that replaces motion annotations with fully or partially annotated (1%, 0.1%) foreground/background masks for supervision. To this end, we develop a weakly supervised approach utilizing foreground/background cues to guide the self-supervised learning of motion prediction models. Since foreground motion generally occurs in non-ground regions, non-ground/ground masks can serve as an alternative to foreground/background masks, further reducing annotation effort. Leveraging non-ground/ground cues, we propose two additional approaches: a weakly supervised method requiring fewer (0.01%) foreground/background annotations, and a self-supervised method without annotations. Furthermore, we design a Robust Consistency-aware Chamfer Distance loss that incorporates multi-frame information and robust penalty functions to suppress outliers in self-supervised learning. Experiments show that our weakly and self-supervised models outperform existing self-supervised counterparts, and our weakly supervised models even rival some supervised ones. This demonstrates that our approaches effectively balance annotation effort and performance.