There is an emerging trend towards the fabrication of microcapsules at liquid interfaces. In order to control the parameters of such capsules, the interfacial processes governing their formation must be understood. Here, poly(vinyl alcohol) films are assembled at the interface of water-in-oil microfluidic droplets. The polymer is cross-linked using cucurbit[8]uril ternary supramolecular complexes. It is shown that compression-induced phase change causes the onset of buckling in the interfacial film. On evaporative compression, the interfacial film both increases in density and thickens, until it reaches a critical density and a phase change occurs. We show that this increase in density can be simply related to the film Poisson ratio and area compression.This description captures fundamentals of many compressive interfacial phase changes and can also explain the observation of a fixed thickness-to-radius ratio at buckling, $(TR)$.