Nonparametric estimation of the transition probability matrix of a progressive multi-state model is consideredunder cross-sectional sampling. Two diﬀerent estimators adapted to possibly right-censored and left-truncated data are pro-posed. The estimators require full retrospective information before the truncation time, which, when exploited, increaseseﬃciency. They are obtained as diﬀerences between two survival functions constructed for sub-samples of subjects occupyingspeciﬁc states at a certain time point. Both estimators correct the oversampling of relatively large survival times by usingthe left-truncation times associated with the cross-sectional observation. Asymptotic results are established, and ﬁnite sampleperformance is investigated through simulations. One of the proposed estimators performs better when there is no censoring,while the second one is strongly recommended with censored data. The new estimators are applied to data on patients inintensive care units (ICUs).