Let $Z$ be a Markov process with state-space $E$. For any finite set $S$ it is possible to associate with $Z$ a process $\zeta$ of coalescing partitions of $S$ with components labelled by elements of $E$ that evolve as copies of $Z$ between coalescences. Subject to very weak hypotheses on $Z$, there is a Feller process $X$ with state-space a certain space of probability measure valued functions on $E$. The process $X$ has its ``moments'' defined in terms of expectations for $\zeta$ in a manner suggested by various instances of martingale problem duality between coalescing Markov processes and voter model particle systems, systems of interacting Fisher-Wright and Fleming-Viot diffusions that arise in population genetics, and stochastic partial differential equations with Fisher-Wright noise that appear as rescaling limits of long-range voter models as well as in population genetics. Some sample path properties are examined in the special case where $Z$ is a symmetric stable process on $\bR$ with index $1 < \alpha \le 2$. In particular, we show that for fixed $t>0$ the essential range of the random probability measure valued function $X_t$ is almost surely a countable set of point masses.