The relative velocity and extra pressure drop of a single vesicle flowing through a square microchannel are quantified via boundary element simulations, lubrication theory and microfluidic experiments. The vesicle is modelled as a fluid sac enclosed by an inextensible, fluidic membrane with a negligible bending stiffness. All results are parametrized in terms of the vesicle sphericity (i.e. the reduced volume) and flow confinement (i.e. the ratio of the vesicle radius to the channel hydraulic radius). Direct comparison is made to previous studies of vesicle flow through circular tubes, revealing several distinct features of the square-channel geometry. Firstly, fluid in the suspending medium bypasses the vesicle through the corners of the channel, which in turn reduces the dissipation created by the vesicle. Secondly, the absence of rotational symmetry about the channel axis permits surface circulation in the membrane (tank treading), which in turn reduces the vesicle’s speed. At very high confinement, both theory and experiment indicate that the vesicle’s speed can be reduced below the mean speed of the suspending fluid through this mechanism. Finally, the contact area for lubrication is greatly reduced in the square-duct geometry, which in turn weakens the stress singularity predicted by lubrication theory. This fact directly leads to a breakdown of the lubrication approximation at low flow confinement, as verified by comparison to boundary element simulations. Since the only distinct property assumed of the membrane is its ability to preserve surface area locally, it is expected that the results of this study are applicable to other types of soft particles with immobilized surfaces (e.g. Pickering droplets, gel beads and biological cells).