We present a novel, physically-based morphing technique for elastic shapes, leveraging the differentiable material point method (MPM) with space-time control through per-particle deformation gradients to accommodate complex topology changes. This approach, grounded in MPM's natural handling of dynamic topologies, is enhanced by a chained iterative optimization technique, allowing for the creation of both succinct and extended morphing sequences that maintain coherence over time. Demonstrated across various challenging scenarios, our method is able to produce detailed elastic deformation and topology transitions, all grounded within our physics-based simulation framework.