Considerable interest has recently been shown in the spatial spread of infection between static individuals. A different approach yielding similar types of processes is to allow the individuals themselves to migrate from site to site. This movement may be physical migration, mutation from one type of organism to another, or it may even represent an approximation to the initial development of a spatial epidemic. We shall consider the population to be distributed amongst either a finite number of sites (model A), or an infinite number situated at the nodes of a lattice (model B). In this latter situation individuals are only allowed to migrate between sites which are nearest neighbours, and the transition rates are assumed to be spatially homogeneous. Individuals at each site undertake a simple birth-death-migration-immigration process.