Thermal convection driven by the kinetic undercooling of solidification can produce compositional and textural variations in an initially uniform lava flow that is cooled predominantly from above. We show that such kinetic undercooling is sufficient to drive convection at high Rayleigh numbers even in shallow lava lakes. We then employ a theoretical model that includes the effects of convection at high Rayleigh number to make predictions of the complete evolution of the lava. We show that predictions of the rate of growth of solid crust at the surface of the lake do not differ much from predictions made by ignoring effects due to convection. However, convective motions do shorten the time for complete solidification because, when they are driven by kinetic undercooling, there is internal crystallization in addition to that occurring near the cooled upper boundary of the lava. This important effect, which is absent from purely conductive models, predicts stratification and zonation of the lava, in agreement with field observations. We also examine the effect of taking into account the heat transfer to the country rock below the lava lake, and determine how the evolution of the lava lake varies with the viscosity of the lava.