The motion skeleton is a core data structure of 3D animation workflows, producing character motions by posing a pre-defined bone hierarchy. Motion data is largely incompatible across skeletons with proportional and/or hierarchical differences, raising long-standing challenges for data-driven motion synthesis. To address this, Temporal Point Clouds (TPC) have emerged as a universal, cross-compatible motion representation, using temporally consistent points that map motion trajectories. While TPCs have demonstrated reversibility with skeletal motions, their role is currently limited to enabling cross-compatibility, whereas we believe motion tasks can be learned directly in the TPC medium. This would require TPC motion synthesis capabilities, which is an unexplored field due to its unique temporal consistency and point identity requirements.In this paper, we propose PUMPS, the primordial auto-encoder architecture for TPC data. It reduces point cloud frames independently into sampleable feature vectors, from which a decoder efficiently extracts distinct temporal points using latent Gaussian noise vectors as sampling identifiers. We introduce linear assignment-based point pairing to optimise the TPC reconstruction process without requiring expensive point-wise attention mechanisms in the architecture. Using the auto-encoder, we produce a pre-trained motion synthesis model capable of performing motion prediction, transition generation, and keyframe interpolation tasks. PUMPS performs remarkably well even without native dataset supervision, matching state-of-the-art performance in its pre-training tasks, and outperforming existing methods when fine-tuned for skeletal motion denoising and estimation tasks.