We investigate crystallization-driven convection in carbon-oxygen white dwarfs. We present a version of the mixing length theory (MLT) that self-consistently includes the effects of thermal diffusion and composition gradients, and provides solutions for the convective parameters based on the local heat and composition fluxes. Our formulation smoothly transitions between the regimes of fast adiabatic convection at large Peclet number and slow thermohaline convection at low Peclet number. It also allows for both thermally-driven and compositionally-driven convection, including correctly accounting for the direction of heat transport for compositionally-driven convection in a thermally-stable background. We use the MESA stellar evolution code to calculate the composition and heat fluxes during crystallization in different models of cooling white dwarfs, and determine the regime of convection and the convective velocity. We find that convection occurs in the regime of slow thermohaline convection during most of the cooling history of the star. However, in more massive white dwarfs the composition flux is large enough at the onset of crystallization to drive fast overturning convection for a short time (). We estimate the convective velocities in both of these phases and discuss the implications for explaining observed white dwarf magnetic fields with crystallization-driven dynamos.