The cut-and-paste process

We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a .-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a Lévy.Itô representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.