A two-dimensional model is proposed to describe systems with phase transitions which take place in terms of crystalline as well as internal degrees of freedom. Computer simulation of the model shows that the interplay between the two sets of degrees of freedom permits observation of grain-boundary formation and interfacial melting, a nonequilibrium process by which the system melts at the boundaries of a polycrystalline domain structure. Lipid membranes are candidates for systems with pronounced interfacial melting behavior.