We use the SPHINX suite of high-resolution cosmological radiation hydrodynamics simulations to study how spatially and temporally inhomogeneous reionization impacts the baryonic content of dwarf galaxies and cosmic filaments. The SPHINX simulations simultaneously capture the large-scale process of reionization, model the escape of ionising radiation from thousands of galaxies, and resolve haloes well below the atomic cooling threshold. This makes them an ideal tool for examining how reionization impacts star formation and the gas content of dwarf galaxies. We compare simulations with and without stellar radiation to isolate the effects of radiation feedback from that of supernova, cosmic expansion, and numerical resolution. We find that the gas content of cosmic filaments can be reduced by more than 80% following reionization. The gas inflow rates into haloes with $Mvir\lesssim 108M\odot$ are strongly affected and are reduced by more than an order of magnitude compared to the simulation without reionization. A significant increase in gas outflow rates is found for halo masses $Mvir\lesssim 7\times 107M\odot$. Our simulations show that inflow suppression (i.e. starvation), rather than photoevaporation, is the dominant mechanism by which the baryonic content of high-redshift dwarf galaxies is regulated. At fixed redshift and halo mass, there is a large scatter in the halo baryon fractions that is entirely dictated by the timing of reionization in the local region surrounding a halo. Finally, although the gas content of high-redshift dwarf galaxies is significantly impacted by reionization, we find that most haloes with $Mvir\lesssim 108M\odot$ can remain self-shielded and form stars long after reionization, until their local gas reservoir is depleted, suggesting that local group dwarf galaxies do not necessarily exhibit star formation histories that peak prior to $z=6$.