We systematically investigate droplet movement, coalescence, and splitting on an open hydrophobic surface. These processes are actuated by magnetic beads internalized in an oil-coated aqueous droplet using an external magnet. Results are organized into an 'operating diagram' that describes regions of droplet stable motion, breakage, and release from the magnet. The results are explained theoretically with a simple model that balances magnetic, friction, and capillary-induced drag forces and includes the effects of particle type, droplet size, surrounding oil layer, surface tension, and viscosity. Finally, we discuss the implications of the results for the design of magnet-actuated droplet systems for applications such as nucleic acid purification, immunoassay and drug delivery.