We study material transport and permeation through a lamellar stack of multi-component lipid membranes by performing Monte Carlo simulations of a stacked two-dimensional Ising model in the presence of permeants. In the model, permeants are transported through the stack via in-plane lipid clusters, which are inter-connected in the vertical direction. These clusters are formed transiently by concentration fluctuations of the lipid mixture, and the permeation process is affected especially close to the critical temperature of the binary mixture. We show that the permeation rate decays exponentially as a function of temperature and permeant lateral size, whereas the dependency on the characteristic waiting time obeys a stretched exponential function. The material transport through such lipid clusters can be significantly affected around physiological temperatures.