Models of multi-agent systems usually update the states of all agents synchronously, but in many real life systems, agents behave asynchronously. Relatively little is yet known about the dynamic characteristics of asynchronous systems. Here we compare synchronous, random asynchronous, and ordered asynchronous updating schemes. Using one-dimensional (1D) cellular automata as a case study, we show that the type of update scheme strongly affects the dynamic characteristics of the system. We also show that global synchronisation can arise from local temporal coupling. Furthermore, it is possible to switch between chaotic, cyclic and modular behaviour by varying a single parameter, which suggests a possible mechanism by which environmental parameters influence emergent structure. We conclude that ordered asynchronous processes with local temporal coupling play a role in self-organisation within many multi-agent systems.