Engineering of atomically thin membranes for hydrogen isotope separation is an actual challenge which has a broad range of applications. Recent experiments [M. Lozada-Hidalgo et al., Science 351, 68 (2016)] unambiguously demonstrate an order-of-magnitude difference in permeabilities of graphene-based membranes to protons and deuterons at ambient conditions, making such materials promising for novel separation technologies. Here we demonstrate that the permeability mechanism in such systems changes from quantum tunneling for protons to quasi-classical transport for heavier isotopes. Quantum nuclear effects exhibit large temperature and mass dependence, modifying the Arrhenius activation energy and Arrhenius prefactor for protons by more than 0.5 eV and by seven orders of magnitude correspondingly. Our findings not only shed light on the separation process for hydrogen isotope ions passing through pristine graphene but also offer new insights for controlling ion transport mechanisms in nanostructured separation membranes by manipulating the shape of the barrier and transport process conditions.