Permeability is a critical property of fractured sorbing rocks, such as shale and coal, as it controls the gas transport. However, two distinctive features are commonly not simultaneously included in existing permeability models. One is the stratified feature which has been evidenced by an appreciable number of laboratory investigations. The second one is matrix-fracture non-equilibrium induced matrix-fracture interactions during permeability measurement or field operations. The goal of this research is to uncover how the combined effects of the two features influence permeability evolution. This study is extended from our previous research (Zeng et al. 2020a) based on the poroelasticity theory. The interbedding layers can be either perpendicular or parallel to the flow direction. Within each layer, the permeability evolution with multiple stages would occur due to the matrix-fracture nonequilibrium and effective stress variation. Therefore, the overall permeability behavior of layered rocks becomes more complex. To investigate the influence of laminae on shale and coal permeability evolution, an equivalent permeability model for multilayer rocks is established by means of weighted-average and weighted-harmonic-average methods. As for matrix-fracture equilibrium hysteresis, the pressure-dependent gas invaded volume ratio concept (Zeng et al. 2020a) is used to describe equilibrium-hysteresis-induced nonuniform matrix swelling. The reliability of this model is checked by comparing with the published model which has been verified against three sets of gas injection experimental data collected from constant effective stress, constant confining pressure, and constant average pore pressure conditions. Our results indicate that the gas flow direction affects the permeability measurement significantly. If the gas flow direction is parallel to the layers, the overall permeability magnitude is close to that of the highest-permeability layer. However, the overall permeability evolution law can be different from that of the highest-permeability layer. When the gas flow is normal to the bedding, both the permeability evolution law and permeability magnitude are similar to those of the lowest-permeability layer.