We study invasion fronts in a class of simple, two-species reaction-diffusion systems that occur as models for recurrent precipitation and undercooled liquids. We exhibit several different modes of front propagation: the invasion of an unstable homogeneous equilibrium can create persistent periodic patterns, transient patterns, or simply a homogeneous state. We give criteria that distinguish between these different modes of invasion, corroborate our predictions with numerical simulations, and point to a rich variety of more subtle phenomena and bifurcations.