While causal mediation analysis has seen considerable recent development for a single measured mediator (M) and final outcome (Y), less attention has been given to repeatedly measured M and Y. Previous methods have typically involved discrete‐time models that limit inference to the particular measurement times used and do not recognize the continuous nature of the mediation process over time. To overcome such limitations, we present a new continuous‐time approach to causal mediation analysis that uses a differential equations model in a potential outcomes framework to describe the causal relationships among model variables over time. A connection between the differential equation models and standard repeated measures models is made to provide convenient model formulation and fitting. A continuous‐time extension of the sequential ignorability assumption allows for identifiable natural direct and indirect effects as functions of time, with estimation based on a two‐step approach to model fitting in conjunction with a continuous‐time mediation formula. Novel features include a measure of an overall mediation effect based on the “area between the curves,” and an approach for predicting the effects of new interventions. Simulation studies show good properties of estimators and the new methodology is applied to data from a cohort study to investigate sugary drink consumption as a mediator of the effect of socioeconomic status on dental caries in children.