The behavior of water, methanol, and water-methanol mixtures confined in narrow slit graphite pores as a function of pore size was investigated by Monte Carlo, hybrid Monte Carlo, and Molecular Dynamics simulations. Interactions were described using TIP4P/2005 for water, OPLS/2016 for methanol, and cross interactions fitted to excess water/methanol properties over the whole range of concentrations, which provide a rather accurate description of water-methanol mixtures. As expected for hydrophobic pores, whereas pure methanol is adsorbed already from the gas phase, pure water only enters the pore at pressures well beyond bulk saturation for all pore sizes considered. When adsorbed from a mixture, however, water adsorbs at much lower pressures due to the formation of hydrogen bonds with previously adsorbed methanol molecules. For all studied compositions and pore sizes, methanol adsorbs preferentially over water at liquid-vapor equilibrium conditions. In pure components, both water and methanol are microscopically structured in layers, the number of layers increasing with pore size. This is also the case in adsorbed mixtures, in which methanol has a higher affinity for the walls. This becomes more evident as the pore widens. Diffusion of pure water is higher than that of pure methanol for all pore sizes due to the larger size of the methyl group. In mixtures, both components present similar diffusivities at all pore sizes, which is explained in terms of the coupling of molecular movements due to strong hydrogen bonding between methanol and water molecules. This is particularly evident in very narrow pores, in which pure methanol diffusion is completely impeded on the time scale of our simulations, but the presence of a small amount of water molecules facilitates alcohol diffusion following a single-file mechanism. Additionally, our results indicate that pure water diffusivities display a non-monotonous dependence of pore size, due to effects of confinement (proximity to a fluid-solid-fluid transition induced by confinement as reported in previous work) and the dynamic anomalies of water. Published under license by AIP Publishing.