The hydrodynamic response of a thin fluid film, whether a Langmuir monolayer at the air/water interface or a cell membrane, is difficult to model, since it involves the coupling of both bulk and surfaces phases. However, such hydrodynamic response is not only intrinsically critical for transport within the layer, it also provides the major available means to evaluate an important parameter for phase-separated layers, the line tension. We have developed a line-integral formulation of the hydrodynamic response of phase-separated layers with short-ranged forces, and tested it by comparisons between numerical simulations based on this model and experiment. These experiments both validate the model and demonstrate that the line tension can be determined with unprecedented accuracy and precision. A simple smectic liquid crystal multilayer, where long-range dipole-dipole interactions are unimportant, is studied in detail. In most monolayers, such interactions are important: as an example, we consider phase coexistence in binary lipid/cholesterol mixed layers.