Contrary to the notion that ‘oil and water do not mix’, many oils possess a residual diffusive mobility through water, causing the drop sizes in oil-in-water emulsions to slowly evolve with time. Liquid interfaces are therefore typically stabilized with polymeric or particulate emulsifiers. Upon adsorption, these may induce strong, localized viscoelasticity in the interfacial region. Here, we show that shrinkage of oil drops due to bulk mass transfer may render such adsorption layers mechanically unstable, causing them to buckle, crumple and, finally, to attain a stationary shape and size. We demonstrate using two types of model interfaces that this only occurs if the adsorption layer has a high interfacial shear elasticity. This is typically the case for adsorbed layers that are cross-linked or ‘jammed’. Conversely, interfacial compression elasticity alone is a poor predictor of interface buckling or arrest. These results provide a new perspective on the role of interfacial rheology for compositional ripening in emulsions. Moreover, they directly affect a variety of applications, including the rapid screening of amphiphilic biopolymers such as the Acacia gum or the octenyl succinic anhydride modified starch used here, the interpretation of light scattering data for size measurements of emulsion drops, or the formulation of delivery systems for encapsulation and release of drugs and volatiles.