Causal mediation analysis is widely used to decompose a treatment's effect into components operating through distinct causal pathways and to characterize the mediator--outcome relationship. In time-to-event settings, competing events --- events that preclude the primary endpoint --- substantially complicate this decomposition. We provide a unified treatment of causal mediation analysis for a time-to-event endpoint under competing events. We show that the total effect on the cause-specific cumulative incidence decomposes into a direct effect, an indirect effect through the mediator of interest, and an indirect effect through the competing events. We develop this decomposition under the natural effects, stochastic interventional effects, and separable effects frameworks, and show that, in certain settings, all three yield identical statistical functionals despite relying on distinct identification assumptions. We derive efficient influence functions for the counterfactual cumulative incidence curves and the decomposed effects, and propose double-machine-learning-type estimators that are multiply robust and attain the semiparametric efficiency bound, with a sensitivity analysis to address unmeasured common causes of time-to-event endpoints. We further study the mediator--outcome response relationship via kernel-smoothed estimands. We apply the methods to immune correlates analyses of COVID-19 vaccine and HIV-1 prevention trials.

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