  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
P\'eclet number, $\mathrm{Pe}$ (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 neutron star and white dwarf cooling. Convection
in neutron stars spans both regimes. We find rapid mixing of neutron star
oceans, with a convective turnover time of order 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 white dwarf magnetic fields.
