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Diffusion processes occurring in a myriad of systems sparkle great interest in understanding their general properties and applications. In this work, we investigate a broad set of diffusive systems that can be governed by a generalized diffusion equation and subjected to a surface that can promote sorption and, consequently, desorption, thus releasing the particles to the bulk. The general bulk equation used here can reproduce different diffusive regimes, among them, those described by the Cattaneo equation or by a fractional, anomalous diffusion. The equation related to the processes on the surface incorporates non-Debye relaxations which can be used to model non-exponential relaxations commonly found in biological or fractal systems. The solutions are obtained by using the Green function approach and show a rich class of behavior that can be related to anomalous diffusion.