A dynamical phase transition from reversible to irreversible behavior occurs when particle suspensions are subjected to uniform oscillatory shear, even in the Stokes flow limit. We consider a more general situation with nonuniform strain (e.g., oscillatory channel flow), which is observed to exhibit markedly different dynamics. The onset of irreversibility is delayed, and occurs simultaneously across the entire channel. This behavior is only partially explained by self-organization and shear-induced migration. The onset is accompanied by long-range correlated particle motion even at the channel center, where the strain is negligible; this motion prevents the system from evolving into a reversible state.