We consider an idealized model of diffusion in a cage. The boundary of the cage can be occupied by various correlated obstacles which fluctuate in time according to a dynamic of a Glauber-type. A particle enters the cage and travels through it according to a given stochastic process. Its exit may be blocked by the presence of the obstacles at the surface of the cage. We show that the probability of transmission through the cage can have a maximum and a minimum as a function of the frequency of fluctuation of the obstacles, provided a certain simple condition on the dynamics inside the cage is satisfied.