Control and understanding of the flow of fluids at nanoscales is of great significance to biology, separation science, energy technology, and medical diagnostics. Nanocarbons have emerged as one of the most promising materials for this quest, both as nanochannels and nanoporous membranes. However, the fluid flow in these graphitic nanostructures is not well understood, and there is a lack of straightforward route for process integration of the nanochannels. The graphene-lined nanochannels (GNCs), reported here, are aimed at solving these problems, while displaying a useful anomaly for fluidic flow. Specifically, GNCs show a large increase in the rate of removal of sacrificial materials enclosed in them. The increase is caused by 100-1000 times enhancement in the diffusivity of etchant media in the GNCs as compared to channels without graphene lining. The enhancement increases monotonically with a decrease in the height of the GNCs, which is not seen for the non-lined channels. These properties, coupled with easy and scalable fabrication, make GNCs highly suited for innovative and efficient nanofluidic devices and also for experimental investigations. We also provide a phenomenological model which assumes enhanced diffusivity of medium only near graphene surface to explain the observed dependence of diffusivity on the dimensions of the nanochannels. This rationalization of the phenomenon using only the surface effects is a significant step towards understanding anomalous fluidics of nanocarbons. © 2016 AIP Publishing LLC.