Sea breezes are often modeled as a wave response to transient heating in a stratified environment. They occur, however, as density currents with well-defined fronts, the understanding of which rests primarily on experiments and theory that do not include the stratification within and above the current and the steady heat input at the land surface. These gaps are investigated here via a sequence of idealized 2D density current simulations, progressing from the simplest classical case to more realistic surface heating and stratification. In the classical situation where the entire horizontal density contrast is imposed initially, the front quickly attains a constant speed determined by traditional formulas based on the density contrast across the front and the current depth, or by the amount of heat needed to produce it from an initially barotropic fluid. However, these diagnostic and prognostic tools fail completely if the current is driven by a gradual input of heat, analogous to a real sea-breeze situation. In this case the current accelerates slowly at first, remaining much slower than would be expected based on classical formulas. The motion of a classical density current is mostly inertial, with accelerations occurring at the current head; while in the continuously heated case, the entire current accelerates, requiring interior body forces to develop slowly owing to heating of the density current itself. This explains why observed sea-breeze fronts propagate more slowly than predicted from classical formulas, and may help to explain why larger landmasses, where fronts have more time to accelerate, often experience stronger convective storms when triggered by sea-breeze effects.