State-of-the-art rigging methods typically assume a predefined canonical rest pose. However, this assumption does not hold for dynamic mesh sequences such as DyMesh or DT4D, where no canonical T-pose is available. When applied independently frame-by-frame, existing methods lack pose invariance and often yield temporally inconsistent topologies. To address this limitation, we propose SPRig, a general fine-tuning framework that enforces cross-frame consistency across a sequence to learn pose-invariant rigs on top of existing models, covering both skeleton and skinning generation. For skeleton generation, we introduce novel consistency regularization in both token space and geometry space. For skinning, we improve temporal stability through an articulation-invariant consistency loss combined with consistency distillation and structural regularization. Extensive experiments show that SPRig achieves superior temporal coherence and significantly reduces artifacts in prior methods, without sacrificing and often even enhancing per-frame static generation quality. The code is available in the supplemental material and will be made publicly available upon publication.