The classical theory of electrodynamics is built upon Maxwells equations and the concepts of electromagnetic field, force, energy and momentum, which are intimately tied together by Poyntings theorem and the Lorentz force law. Whereas Maxwells macroscopic equations relate the electric and magnetic fields to their material sources (i.e., charge, current, polarization and magnetization), Poyntings theorem governs the flow of electromagnetic energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and the fields. The close association of momentum with energy thus demands that the Poynting theorem and the Lorentz law remain consistent with each other, while, at the same time, ensuring compliance with the conservation laws of energy, linear momentum, and angular momentum. This paper shows how a consistent application of the aforementioned laws of electrodynamics to moving permanent dipoles (both electric and magnetic) brings into play the rest-mass of the dipoles. The rest mass must vary in response to external electromagnetic fields if the overall energy of the system is to be conserved. The physical basis for the inferred variations of the rest-mass appears to be an interference between the internal fields of the dipoles and the externally applied fields. We use two different formulations of the classical theory in which energy and momentum relate differently to the fields, yet we find identical behavior for the restmass in both formulations.