Summary: It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman’s two-parameter Chinese restaurant process, tree-growth models associated with Mallows’ \(\phi\) model of random permutations and with M. P. Schützenberger’s [C. R. Acad. Sci., Paris 236, 352–353 (1953; Zbl 0051.09401)] non-commutative \(q\)-binomial theorem, and a construction due to M. Luczak and P. Winkler [Random Struct. Algorithms 24, No. 4, 420–443 (2004; Zbl 1050.60007)] that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail \(\sigma\)-fields.