We investigate a contaminant transport in fractal media with randomly inhomogeneous diffusion barrier. The diffusion barrier is a low-permeable matrix with rare high-permeability pathways (punctures). At times, less than a characteristic matrix diffusion time, the problem is effectively barrier-free, where an effective source acts during the time t(eff) << t. The punctures result in a precursor contaminant concentration at short times and additional stage of the asymptotic concentration distribution at long times. If the source surface area is large enough, then the barrier can be considered as statistical homogeneous medium; otherwise, strong fluctuations occur.