We consider the problem of quickest detection of dynamically evolving events in sensor networks. After an event occurs, a number of sensors are affected and undergo a change in the statistics of their observations. We assume that the event is dynamic and can propagate with time, i.e., different sensors perceive the event at different times. The goal is to design a sequential test that can detect when the event has affected no fewer than a certain pre-dertermined number of sensors as quickly as possible, subject to false alarm constraints. We design a computationally efficient algorithm for this problem that is adaptive to the unknown propagation dynamics, and demonstrate its asymptotic optimality as the false alarm rate goes to zero. We also provide numerical simulations to validate our theoretical results.