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 $\sigma$-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.
"The cut-and-paste process." Ann. Probab. 42 (5) 1952 - 1979, September 2014. https://doi.org/10.1214/14-AOP922