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Longitudinal modified treatment policies (LMTP) have been recently developed as a novel method to define and estimate causal parameters that depend on the natural value of treatment. LMTPs represent an important advancement in causal inference for longitudinal studies as they allow the non-parametric definition and estimation of the joint effect of multiple categorical, numerical, or continuous exposures measured at several time points. We extend the LMTP methodology to problems in which the outcome is a time-to-event variable subject to right-censoring and competing risks. We present identification results and non-parametric locally efficient estimators that use flexible data-adaptive regression techniques to alleviate model misspecification bias, while retaining important asymptotic properties such as $\sqrt{n}$-consistency. We present an application to the estimation of the effect of the time-to-intubation on acute kidney injury amongst COVID-19 hospitalized patients, where death by other causes is taken to be the competing event.