A colloidal particle placed in an inhomogeneous solution of smaller nonadsorbing polymers will move towards regions of lower polymer concentration in order to reduce the free energy of the interface between the surface of the particle and the solution. This phenomenon is known as diffusiophoresis. Treating the polymer as penetrable hard spheres, as in the Asakura-Oosawa model, a simple analytic expression for the diffusiophoretic drift velocity can be obtained. In the context of drying films we show that diffusiophoresis by this mechanism can lead to stratification under easily accessible experimental conditions. By stratification we mean spontaneous formation of a layer of polymer on top of a layer of the colloid. Transposed to the case of binary colloidal mixtures, this offers an explanation for the stratification observed recently in these systems [A. Fortini et al., Phys. Rev. Lett. 116, 118301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.118301]. Our results emphasize the importance of treating solvent dynamics explicitly in these problems and caution against the neglect of hydrodynamic interactions or the use of implicit solvent models in which the absence of solvent backflow results in an unbalanced osmotic force that gives rise to large but unphysical effects.