Various classes of infinite-state processes are often specified by rewrite systems. We extend Mayr's Process Rewrite Systems (PRS) [Information and Computation 156 (2000) 264–286] by finite-state unit whose transition function satisfies some restrictions inspired by weak finite automata. We classify these models by their expressiveness and show how the hierarchy of new classes (w.r.t. bisimilarity) is related to both PRS hierarchy of Mayr and two other hierarchies of PRS extensions introduced in [Theoretical Computer Science 258 (2001) 409–433; Electronic Notes in Theoretical Computer Science 52 (2002)].