In this article, we will present statistical methods to assess to what extent the effect of a randomised treatment (versus control) on a time-to-event endpoint might be explained by the effect of treatment on a mediator of interest, a variable that is measured longitudinally at planned visits throughout the trial. In particular, we will show how to identify and infer the path-specific effect of treatment on the event time via the repeatedly measured mediator levels. The considered proposal addresses complications due to patients dying before the mediator is assessed, due to the mediator being repeatedly measured, and due to posttreatment confounding of the effect of the mediator by other mediators. We illustrate the method by an application to data from the LEADER cardiovascular outcomes trial.