We generalize the recently developed "internal" density-functional theory (DFT) and Kohn-Sham scheme to multicomponent systems. We obtain a general formalism, applicable for the description of multicomponent self-bound systems (such as molecular systems where the nuclei are treated explicitly, atomic nuclei and mixtures of 3He and 4He droplets), where the fundamental translational symmetry has been treated correctly. The main difference with traditional DFT is the explicit inclusion of center-of-mass correlations in the functional. A large part of the paper is dedicated to the application to molecular systems, which permits us to clarify the approximations that underly traditional DFT.