A Lagrangian framework for material transport along hydrological pathways is presented and consequences of statistically stationary space-time flow velocity variations on advective transport are investigated. The two specific questions addressed in this work are: How do temporal fluctuations affect forward and backward water travel time distributions when combined with spatial variability? and Can mass transfer processes be quantified using conditional probabilities in spatially and temporally variable flow? Space-time trajectories are studied for generic conditions of flow, with fully ergodic or only spatially ergodic velocity. It is shown that forward and backward distributions of advective water travel time coincide for statistically stationary space-time variations. Temporal variability alters the statistical structure of the Lagrangian velocity fluctuations. Once this is accounted for, integration of the memory function with the travel time distribution is applicable for quantifying retention. Further work is needed to better understand the statistical structure of space-time velocity variability in hydrological transport, as well as its impact on tracer retention and attenuation.