In this paper we consider non-local (in time) heat equations on time-increasing parabolic sets whose boundary is determined by a suitable curve. We provide a notion of solution for these equations and we study well-posedness under Dirichlet conditions outside the domain. A maximum principle is proved and used to derive uniqueness and continuity with respect to the initial datum of the solutions of the Dirichlet problem. Existence is proved by showing a stochastic representation based on the delayed Brownian motion killed on the boundary. Several related distributional properties of the delayed Brownian motion and its crossing probabilities are also obtained. The asymptotic behaviour of the mean square displacement of the process is determined, showing that the diffusive behaviour is anomalous.