  We develop a scheme for the computation of the full-counting statistics of
transport described by Markovian master equations with an arbitrary time
dependence. It is based on a hierarchy of generalized density operators, where
the trace of each operator yields one cumulant. This direct relation offers a
better numerical efficiency than the equivalent number-resolved master
equation. The proposed method is particularly useful for conductors with an
elaborate time-dependence stemming, e.g., from pulses or combinations of slow
and fast parameter switching. As a test bench for the evaluation of the
numerical stability, we consider time-independent problems for which the
full-counting statistics can be computed by other means. As applications, we
study cumulants of higher order for two time-dependent transport problems of
recent interest, namely steady-state coherent transfer by adiabatic passage and
Landau-Zener-St\"uckelberg-Majorana interference in an open double quantum dot.
