Radiation from magnetic and electric dipole moments is a key subject in the theory of electrodynamics. Although people treat the problem thoroughly in the context of the frequency domain, the problem is still not well understood in the context of the time domain especially if dipole moments arbitrarily vary in time under the action of external forces. Here, we scrutinize the instantaneous power radiated by magnetic and electric dipole moments and report findings that are different from the conventional understanding of their instantaneous radiation found in textbooks. In contrast to the traditional far-field approach based on the Poynting vector, our analysis employs a near-field method based on the induced electromotive force, leading to corrective terms that are found to be consistent with time-domain numerical simulations, unlike previously reported expressions. Beyond its theoretical value, this paper may also have significant impact in the field of time-varying metamaterials especially in the study of radiation from subwavelength meta-atoms, scatterers, and emitters that are temporally modulated.