  We study a family of Markov processes on $\mathcal{P}^{(k)}$, the space of
partitions of the natural numbers with at most $k$ blocks. The process can be
constructed from a Poisson point process on
$\mathbb{R}^+\times\prod_{i=1}^k\mathcal{P}^{(k)}$ with intensity
$dt\otimes\varrho_{\nu}^{(k)}$, where $\varrho_{\nu}$ is the distribution of
the paintbox based on the probability measure $\nu$ on $\masspartition$, the
set of ranked-mass partitions of 1, and $\varrho_{\nu}^{(k)}$ is the product
measure on $\prod_{i=1}^k\mathcal{P}^{(k)}$. 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.
