Population genetics keywords, in approximate order of generality and chronology:
Fisher-Wright diffusion, Moran process, Viot-Fleming process.
Obviously there are many other processes matching this broad description.
For one example, any element-wise-positive vector-valued stochastic process can be
made into a discrete-measure-valued process by normalising the state vector to
sum to 1.
For another, count time series are realisations of the measures of such as process of these.
Chinese-restaurant processes as processes in time presumably fit here, although
AFAICT the use of these processes in the literature is usually not the
time-evolving construction, but rather the infinite-time limit of such a process,
which is confusing nomenclature.
If the process does not take values in discrete measures, that would be a
different notebook, which does not at the moment exist;
The Dirichlet process, despite its name is not usually sampled as a time process
and in any case has boring dynamics. However, it does I believe have
time-indexed extensions (Griffin and Steel 2006; Dunson and Park 2008)
Asselah, Amine, Pablo A. Ferrari, and Pablo Groisman. 2011. “Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces.”Journal of Applied Probability 48 (2): 322–32. https://doi.org/10.1239/jap/1308662630.
Donnelly, Peter, and Thomas G. Kurtz. 1996. “A Countable Representation of the Fleming-Viot Measure-Valued Diffusion.”The Annals of Probability 24 (2): 698–742. http://www.jstor.org/stable/2244946.
Ethier, S. N., and Thomas G. Kurtz. 1993. “Fleming–Viot Processes in Population Genetics.”SIAM Journal on Control and Optimization 31 (2): 345–86. https://doi.org/10.1137/0331019.
Fleming, Wendell H, and Michel Viot. 1979. “Some Measure-Valued Markov Processes in Population Genetics Theory.”Indiana University Mathematics Journal 28 (5): 817–43. http://www.jstor.org/stable/24892583.