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Selective solvent and solute transport across nanopores is fundamental to membrane separations, yet it remains poorly understood, especially for non-aqueous systems. Here, we design a chemically robust nanoporous graphene membrane and study molecular transport in various organic liquids under subnanometre confinement. We show that the nature of the solvent can modulate solute diffusion across graphene nanopores, and that breakdown of continuum flow occurs when pore size approaches the solvent's smallest molecular cross-section. By holistically engineering membrane support, modelling pore creation and defect management, high rejection and ultrafast organic solvent nanofiltration of dye molecules and separation of hexane isomers are achieved. The membranes exhibit stable fluxes across a range of solvents, consistent with flow across rigid pores whose size is independent of the solvent. These results demonstrate that nanoporous graphene is a rich materials system for controlling subcontinuum flow that could enable new membranes for a range of challenging separation needs.