Balancing extensive quantities at interfaces in terms of excess densities is a subtle problem because these densities change with the precise location of the dividing surface within the interfacial region. We propose to handle such ambiguities for moving interfaces by introducing a gauge degree of freedom associated with the location of the dividing surface and by studying all the normal velocities associated with different excess densities. Unambiguous interfacial balance equations can then be obtained directly from the differences between normal velocities. By assuming local equilibrium, considering the interfacial entropy balance, and identifying the entropy production rate at the interface, we are naturally led to thermodynamically consistent constitutive assumptions for the fluxes characterizing transport in the interface and for the boundary conditions characterizing transport across the interface. The usefulness and generality of the proposed approach is illustrated in the context of several examples. (c) 2014 American Institute of Chemical Engineers AIChE J, 60: 1424-1433, 2014