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The Aalen-Johansen estimator generalizes the Kaplan-Meier estimator for independently left-truncated and right-censored survival data to estimating the transition probability matrix of a time-inhomogeneous Markov model with finite state space. Such multi-state models have a wide range of applications for modelling complex courses of a disease over the course of time, but the Markov assumption may often be in doubt. If censoring is entirely unrelated to the multi-state data, it has been noted that the Aalen-Johansen estimator, standardized by the initial empirical distribution of the multi-state model, still consistently estimates the state occupation probabilities. Recently, this result has been extended to transition probabilities using landmarking, which is, inter alia, useful for dynamic prediction. We complement these results in three ways. Firstly, delayed study entry is a common phenomenon in observational studies, and we extend the earlier results to multi-state data also subject to left-truncation. Secondly, we present a rigorous proof of consistency of the Aalen-Johansen estimator for state occupation probabilities, on which also correctness of the landmarking approach hinges, correcting, simplifying and extending the earlier result. Thirdly, our rigorous proof motivates wild bootstrap resampling. Our developments for left-truncation are motivated by a prospective observational study on the occurrence and the impact of a multi-resistant infectious organism in patients undergoing surgery. Both the real data example and simulation studies are presented. Studying wild bootstrap is motivated by the fact that, unlike drawing with replacement from the data, it is desirable to have a technique that works both with non-Markov models subject to random left-truncation and right-censoring and with Markov models where left-truncation and right-censoring need not be entirely random.