In a recent paper, we reported a particular class of dynamic oscillatory instabilities induced by the time-dependent adsorption of surfactant along the wetting meniscus of the concave oil-water-solid interface. A rather phenomenological description of these instabilities was then proposed. In the present paper, a quite different configuration consisting of a convex wetting meniscus is used to check for the dependence of this oscillatory phenomenon on the geometry and direction of the triple phase line (TPL) motion as well. As for the concave wetting meniscus, it is found that the adsorption along the interfaces still results in oscillating instabilities as the meniscus readjusts its capillary parameters. These instabilities are essentially related and described through dynamic fluctuations of the surface concentration and hence surface pressure along the self-assembling film at the mobile oil-water interface. Based on a cyclic compression-relaxation process of the monolayer, a comprehensive picture is proposed for these surfactant-mediated instabilities. This involves both molecular exchange mechanisms between the metastable film and the adjacent subphases and molecular reorganization within the monolayer.