Abstract Self-activation coupled to a transport mechanism results in traveling waves that describe polymerization reactions, forest fires, tumor growth, and even the spread of epidemics. Diffusion is a simple and commonly used model of particle transport. Many physical and biological systems are, however, better described by persistent random walks that switch between multiple states of ballistic motion. So far, traveling fronts in persistent random walk models have only been analyzed in special, simplified cases. Here, we formulate the general model of reaction-transport processes in such systems and show how to compute the expansion velocity for arbitrary number of states. For the two-state model, we obtain a closed-form expression for the velocity and report how it is affected by different transport and replication parameters. We also show that nonzero death rates result in a discontinuous transition from quiescence to propagation. We compare our results to a recent observation of a discontinuous onset of propagation in microtubule asters and comment on the universal nature of the underlying mechanism.