A localized insoluble monolayer of nonuniform concentration on the free surface of a thin viscous fluid layer will, through the generation of surface-tension gradients, drive a shear flow causing the monolayer to spread and inducing large deformations of the free surface of the liquid layer. An existing model of surfactant-driven flows, based on lubrication theory, is extended here to incorporate the transport of a passive solute and absorption of the solute at the wall beneath the liquid layer. A combination of numerical and averaging techniques are employed, appropriate to a wide range of solute diffusivities, time scales and absorption rates. It is shown that diffusion of the solute across the liquid layer causes the solute to be advected on average more slowly than the surfactant, because the solute is carried into regions moving more slowly than the fluid at the free surface; diffusion along the layer may overwhelm advection in certain circumstances, however. Because of the flow-induced deformation of the liquid layer depth, absorption at the lower boundary, if it occurs, does so in a spatially nonuniform manner. The results have implications for techniques of pulmonary drug delivery and flow visualization.