Instability driven by diffusion flux across an asymmetric liquid film is theoretically investigated. To specify the system, we consider an oil-water-air film; the transported species are oil molecules. The latter are dissolved (solubilized) at the oil-water interface, then they are transferred by diffusion across the aqueous film to the water-air interface, where they penetrate between the tails of adsorbed surfactant; molecules. Fluctuational capillary waves at the two film surfaces are considered. The set of theoretical equations is solved to derive a dispersion relation between the wavenumber and the exponent of wave growth. The results reveal that if a local decrease in the film thickness appears, the water-air interface enters a zone enriched in dissolved oil. The newly adsorbed oil creates a surface tension gradient, which carries water away and causes a further decrease of the local film thickness in the concave zone, until eventually the film ruptures. The numerical results show that the film becomes less stable when its thickness decreases, its radius increases, and the diffusion flux across it is more intensive. Even very small decrements of the water-air surface tension, caused by the adsorbed oil, are sufficient to trigger the instability. Its appearance is not so sensitive to the degree of mobility of the oil-water interface. If the water-air surface is preequilibrated with the transported component (there is prespread oil), then the surface tension decrement and the instability disappear. The rupture of an asymmetric oil-water-air film is a precondition for entry of an oil drop at the water-air interface and effectuation of its foam-destructive action. Our results about a specific source of film instability could be helpful for a deeper understanding of the mechanisms of antifoaming.