Electrokinetics in salt-free media (in which counterions are only present) is central to the performance of many systems of modern technological relevance, ranging from ion-selective nanopores to electronic papers. Here, we introduce an analytical theory to describe the size dependence of electroosmosis in such typical scenarios, exhibiting an interesting confluence of the implications of interdependence of the electroosmotic transport mechanisms, ionic sizes, and confinement dimensions along with the counterion concentration. Our results do reveal that the concerned mobility parameter, describing the strength of electroosmotic transport, increases simultaneously with increments in the surface charge density as well as an ionic size factor (also known as the steric factor), bearing far-ranging consequences in microfluidic and nanofluidic technology.