Brains are composed of connected neurons that compute by transmitting signals. The neurons are generally fixed in space, but the communication patterns that enable information processing change rapidly. By contrast, other biological systems, such as ant colonies, bacterial colonies, slime moulds and immune systems, process information using agents that communicate locally while moving through physical space. We refer to systems in which agents are strongly connected and immobile as solid, and to systems in which agents are not hardwired to each other and can move freely as liquid. We ask how collective computation depends on agent movement. A liquid cellular automaton (LCA) demonstrates the effect of movement and communication locality on consensus problems. A simple mathematical model predicts how these properties of the LCA affect how quickly information propagates through the system. While solid brains allow complex network structures to move information over long distances, mobility provides an alternative way for agents to transport information when long-range connectivity is expensive or infeasible. Our results show how simple mobile agents solve global information processing tasks more effectively than similar systems that are stationary.This article is part of the theme issue ‘Liquid Brains, Solid Brains’.