Engineering synthetic materials that mimic the remarkable complexity of living organisms is a fundamental challenge in science and technology. We studied the spatiotemporal patterns that emerge when an active nematic film of microtubules and molecular motors is encapsulated within a shape-changing lipid vesicle. Unlike in equilibrium systems, where defects are largely static structures, in active nematics defects move spontaneously and can be described as self-propelled particles. The combination of activity, topological constraints, and vesicle deformability produces a myriad of dynamical states. We highlight two dynamical modes: a tunable periodic state that oscillates between two defect configurations, and shape-changing vesicles with streaming filopodia-like protrusions. These results demonstrate how biomimetic materials can be obtained when topological constraints are used to control the non-equilibrium dynamics of active matter.