We consider some classes of stationary, counting--measure--valued Markov processes and their companions under time--reversal. Examples arise in the L\'evy--It\^o decomposition of stable Ornstein--Uhlenbeck processes, the large--time asymptotics of the standard additive coalescent, and extreme value theory. These processes share the common feature that points in the support of the evolving counting--measure are born or die randomly, but each point follows a deterministic flow during its life--time.