We study a family of Markov processes on P
(k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R
+ x ∏
i=1
k
P
(k) with intensity dt ⊗ ϱν
(k), where ϱν is the distribution of the paintbox based on the probability measure ν on P
m, the set of ranked-mass partitions of 1, and ϱν
(k) is the product measure on ∏
i=1
k
P
(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.