Spatially uniform magnetic fields induce nonzero forces on magnetic particles adsorbed at curved liquid interfaces thereby driving their motion. Such motions, prohibited in bulk fluids, arise due to interfacial constraints that couple magnetic torques to capillary forces at curved interfaces. Here, we show that time-varying (spatially uniform) magnetic fields can be used to drive a variety of steady particle motions on water drops in decane. Upon application of a precessing field, magnetic Janus particles with amphiphilic surface chemistry move either along circular orbits at the drop poles or along zigzag paths at the drop equator. The different magneto-capillary motions depend on the frequency and precession angle of the field as well as the initial position of the particle on the drop surface. Our experimental observations are reproduced and explained by a mathematical model that accounts for the relevant magnetic, capillary, and hydrodynamic forces and torques that contribute to particle motion. In addition to ferromagnetic Janus particles, we show that similar dynamics can be achieved using superparamagnetic carbonyl iron particles, which are manufactured on industrial scales and respond to even weaker magnetic fields. We demonstrate how the field-driven motion of such particles at the drop interface can induce fluid flows that effectively mix the drop interior. These results suggest that magneto-capillary particle motions could be used to enhance mass transfer within emulsions stabilized by magnetic particles.