Fish schools and bird flocks are fascinating examples of collective behaviours in which many individuals generate and interact with complex flows. Motivated by animal groups on the move, here we explore how the locomotion of many bodies emerges from their flow-mediated interactions. Through experiments and simulations of arrays of flapping wings that propel within a collective wake, we discover distinct modes characterized by the group swimming speed and the spatial phase shift between trajectories of neighbouring wings. For identical flapping motions, slow and fast modes coexist and correspond to constructive and destructive wing–wake interactions. Simulations show that swimming in a group can enhance speed and save power, and we capture the key phenomena in a mathematical model based on memory or the storage and recollection of information in the flow field. These results also show that fluid dynamic interactions alone are sufficient to generate coherent collective locomotion, and thus might suggest new ways to characterize the role of flows in animal groups.