We investigate the migration by thermal regelation of single particles and clusters of particles surrounded by ice subjected to a temperature gradient. This phenomenon is relevant to the casting of porous materials, to cryopreservation of biological tissue, and to the degradation of paleoclimatic signals held in ice sheets, for example. Using carefully controlled laboratory experiments, we measure the migration rates of single particles and clusters as they approach the freezing front. We find that clusters migrate at a constant rate, while single particles accelerate towards the freezing front. This fundamental difference is attributed to the fact that, during regelation, melt water passes through the interstices of a cluster, limited by its constant permeability, but for a single particle must flow through a thin layer of pre-melted ice whose thickness diverges as the freezing temperature is approached, reducing the viscous resistance to migration. We extend existing theories of particle and cluster migration to include the influences of different thermal conductivities and of latent heat on the local temperature field in and around the particle or cluster. We find that if the specific latent heat is large or the viscous resistance to flow is sufficiently small then the migration rate is determined solely by heat transport.