Sea breezes and gust front outflows are thought to behave as classical density currents. However, the understanding of such currents rests primarily on experiments and theory that do not address certain characteristics of real atmospheric flows—in particular, continuous stratification within and above the current, and steady input of heat at the surface as the flow develops. Past studies are extended here by conducting several idealized 2-D simulations progressing from the simplest classical case to ones with more realistic surface heating and stratification. Results reveal that in the classical situation where the entire density contrast is imposed initially, the propagation speed of the density current front quickly attains a constant value that is well diagnosed by traditional formulas based on the observed height and density contrast across the front. Moreover, the speed can be well predicted from the initial condition, over a wide range of stratifications, based on the amount of heat needed to produce it from an initially barotropic fluid. However, these diagnostic and prognostic tools fail completely in the case where the current is accelerated by a gradual input of heat, analogous to a real sea-breeze situation. In this case the current accelerates very slowly at first, remaining much slower than would be expected based on classical formulas, finally attaining the predicted speed only several hours after heating is switched off. The classical formulas succeed in the classical case because the motion is mostly inertial, with accelerations occurring only at the current head; in the continuously heated case, however, the entire current must accelerate. The interior body forces required for this develop slowly as a result of the heating of the density current itself. This explains why observed sea-breeze fronts propagate more slowly that predicted from classical formulas, and may help to explain why larger land masses, where fronts have more time to accelerate, often experience stronger convective storms when triggered by sea-breeze effects.