Randomized fault-tolerant distributed algorithms pose a number of challenges for automated verification: (i) parameterization in the number of processes and faults, (ii) randomized choices and probabilistic properties, and (iii) an unbounded number of asynchronous rounds. The combination of these challenges makes verification hard. Challenge (i) was recently addressed in the framework of threshold automata. We extend threshold automata to model randomized algorithms that perform an unbounded number of asynchronous rounds. For non-probabilistic properties, we show that it is necessary and sufficient to verify these properties under round-rigid schedules, that is, schedules where no process enters round r before all other processes finished round r −1. For almost-sure termination, we analyze these algorithms under round-rigid adversaries , that is, fair adversaries that only generate round-rigid schedules. This allows us to do compositional and inductive reasoning that reduces verification of the asynchronous multi-round algorithms to model checking of a one-round threshold automaton. We apply this framework to classic algorithms: Ben-Or's and Bracha's seminal consensus algorithms for crashes and Byzantine faults, 2-set agreement for crash faults, and RS-Bosco for the Byzantine case.