Age-structured processes are well-established in population biology, where birth and death rates often depend on the age of the underlying populations. Recently, however, different examples of age-structured processes have been considered in the context of cell motility or certain types of stochastically gated ion channels, where the state of the system is determined by a switching process with age-dependent transition rates. In this paper we consider the particular problem of diffusion on a finite interval, with randomly switching boundary conditions due to the presence of an age-structured stochastic gate at one end of the interval. When the gate is closed the particles are reflected, whereas when it is open the domain is in contact with a particle bath. We use a moments method to derive a partial differential equation for the expectations of the stochastic concentration, conditioned on the state of the gate. We then use transform methods to eliminate the residence time of the age-structured switching, resulting in non-Markovian equations for the expectations, and determine the effective steady-state concentration gradient. Our analytical results are shown to match those obtained using Monte Carlo simulations.