A kinetic treatment of the transport of mobile molecules in nanometer channels is presented, that explores ideas of the Enskog theory of dense gases combined with molecular relaxation to the substrate. Transport diffusivity is shown to increase strongly with density, as an effect of effective transfer of mass in the nonlocal intermolecular collisions. The effect is partially compensated by the additional collisional relaxation, that would be absent in the truly one-dimensional system of hard rods, but is shown to occur in the realistic cases of quasi-one-dimensional channels and/or ''soft'' particles. A resulting behavior of diffusivity as a function of density close to 1/(1-theta), where theta is the degree of occupancy, is expected. This suggests an explanation for the puzzling universal behavior observed earlier in zeolites. (C) 1997 American Institute of Physics.