In order to account for diffusio-osmosis, Derjaguin proposed long ago that there is an excess pressure confined within a layer of typically a few nanometers in the vicinity of a solid surface immersed in a liquid and resulting from the interaction between the liquid and the surface. In the presence of a composition gradient in the liquid a confined pressure gradient parallel to the surface is therefore responsible for the diffusio-osmotic flow. This picture appears in contradiction with the contact theorem of colloidal science according to which such excess pressure does not exist. We propose a theoretical description for calculating hydrodynamic flows in inhomogeneous liquids in the vicinity of solid interfaces which is consistent with the contact theorem. This approach is based on a Gibbs free energy and a virtual work principle for calculating the driving forces in the liquid due to inhomogeneous composition along a capillary and to the interaction with the solid interfaces. Our approach allows us to show that the physics at play is the same in wetting or in diffusio-osmosis experiments, as one can go continuously from the latter to the former by making composition gradients sharper. We obtain an explicit expression for the diffusio-osmotic mobility which depends on the Gibbs free energy density in the vicinity of the interface and its dependance on the solute concentration in the liquid beyond the interfacial region, and which is inversely proportional to the liquid viscosity.