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We use the EAGLE suite of hydrodynamical simulations to analyse accretion rates (and the breakdown of their constituent channels) onto haloes over cosmic time, comparing the behaviour of baryons and dark matter (DM). We also investigate the influence of sub-grid baryon physics on halo-scale inflow, specifically the consequences of modelling radiative cooling, as well as feedback from stars and active galactic nuclei (AGN). We find that variations in halo baryon fractions at fixed mass (particularly their circum-galactic medium gas content) are very well correlated with variations in the baryon fraction of accreting matter, which we show to be heavily suppressed by stellar feedback in low-mass haloes, $M_{\rm halo}\lesssim10^{11.5}M_{\odot}$. Breaking down accretion rates into first infall, recycled, transfer and merger components, we show that baryons are much more likely to be smoothly accreted than to have originated from mergers when compared to DM, finding (averaged across halo mass) a merger contribution of $\approx6\%$ for baryons, and $\approx15\%$ for DM at $z\approx0$. We also show that the breakdown of inflow into different channels is strongly dependent on sub-grid physics, particularly the contribution of recycled accretion (accreting matter that has been previously ejected from progenitor haloes). Our findings highlight the dual role that baryonic feedback plays in regulating the evolution of galaxies and haloes: by (i) directly removing gas from haloes, and (ii) suppressing gas inflow to haloes.