Epoxy membranes with same chemical structure but different free volumes were prepared using aliphatic epoxide and polyether diamines as cross-linking agents. Their fractional free volume f(h)(T) was previously evaluated as a function of temperature and cross-linking density [P. N. Patil et al., PhysChemChemPhys 18 (2016) 3817-3824]. Original H-2 permeation measurements were performed changing sample temperature and network structure and compared with CO2 and N-2 transport data. Modelling gas permeability Phi in the framework of the free volume theory we observe that the Arrhenius-type behavior of Phi for test gases is consequence of the f(h)(T) linear increase with temperature and that at equal f(h) value membranes exhibit increasing Phi values by decreasing the cross-linking density. Analysis of permeability data on this set of test molecules with different size and condensation properties suggests that increasing the cross-linking between polymer chains limits their thermal fluctuations and the redistribution of the free volume elements where diffusional jumps of penetrant molecules occur. (C) 2017 Elsevier Ltd. All rights reserved.