Domain migration is observed on the surface of ternary giant unilamellar vesicles held in a temperature gradient in conditions where they exhibit coexistence of two liquid phases. The migration localizes domains to the hot side of the vesicle, regardless of whether the domain is composed of the more ordered or disordered phase and regardless of the proximity to chamber boundaries. The distribution of domains is explored for domains that coarsen and for those held apart due to long-range repulsions. After considering several potential mechanisms for the migration, including the temperature preferences for each lipid, the favored curvature for each phase, and the thermophoretic flow around the vesicle, we show that observations are consistent with the general process of minimizing the system's line tension energy, because of the lowering of line interface energy closer to mixing. DNA strands, attached to the lipid bilayer with cholesterol anchors, act as an exemplar "cargo," demonstrating that the directed motion of domains toward higher temperatures provides a route to relocate species that preferentially reside in the domains.