We study the dynamics of topological bound states after a proximity-coupled to a topologically trivial, gapless medium. Remarkably, we find that a number of properties that characterize the topological nature of these bound states, such as fractional charge and statistics, are dynamically induced in the gapless medium. We show that this dynamically induced topology can be understood in terms of the propagation of entanglement between the topological and gapless systems. We also characterize the effects of interactions and disorder in the medium on the integrity of the topological bound states and the ensuing dynamics.