Mediation analysis is a useful tool to illuminate the mechanisms through which an exposure affects an outcome but statistical challenges exist with time-to-event outcomes and longitudinal observational data. Natural direct and indirect effects cannot be identified when there are exposure-induced confounders of the mediator-outcome relationship. Previous measurements of a repeatedly-measured mediator may themselves confound the relationship between the mediator and the outcome. To overcome these obstacles, two recent methods have been proposed, one based on path-specific effects and one based on an additive hazards model and the concept of exposure splitting. We investigate these techniques, focusing on their application to observational datasets. We apply both methods to an analysis of the UK Cystic Fibrosis Registry dataset to identify how much of the relationship between onset of cystic fibrosis-related diabetes and subsequent survival acts through pulmonary function. Statistical properties of the methods are investigated using simulation. Both methods produce unbiased estimates of indirect and direct effects in scenarios consistent with their stated assumptions but, if the data are measured infrequently, estimates may be biased. Findings are used to highlight considerations in the interpretation of the observational data analysis.