We investigate heat transport associated with compositionally driven convection driven by crystallization at the ocean–crust interface in accreting neutron stars, or growth of the solid core in cooling white dwarfs. We study the effect of thermal diffusion and rapid rotation on the convective heat transport, using both mixing length theory and numerical simulations of Boussinesq convection. We determine the heat flux, composition gradient, and Péclet number, Pe (the ratio of thermal diffusion time to convective turnover time) as a function of the composition flux. We find two regimes of convection with a rapid transition between them as the composition flux increases. At small Pe, the ratio between the heat flux and composition flux is independent of Pe, because the loss of heat from convecting fluid elements due to thermal diffusion is offset by the smaller composition gradient needed to overcome the reduced thermal buoyancy. At large Pe, the temperature gradient approaches the adiabatic gradient, saturating the heat flux. We discuss the implications for cooling of neutron stars and white dwarfs. Convection in neutron stars spans both regimes. We find rapid mixing of neutron star oceans, with a convective turnover time of the order of weeks to minutes depending on rotation. Except during the early stages of core crystallization, white dwarf convection is in the thermal-diffusion-dominated fingering regime. We find convective velocities much smaller than recent estimates for crystallization-driven dynamos. The small fraction of energy carried as kinetic energy calls into question the effectiveness of crystallization-driven dynamos as an explanation for observed magnetic fields in white dwarfs.