Abstract Surface gravity driven currents of varying scales play a major role in both natural and human-made settings. The driving buoyancy forces for these flows are due to horizontal density gradients that may be due to compositional differences across an interface arising from salinity or temperature contrasts. These density gradients may also arise due to the presence of suspended solid material in the current as in the case of turbidity currents, pyroclastic flows or powder snow avalanches. In this article we will model a class of surface gravity flows wherein the density gradients are due to surface fluxes of both heat and salinity. With the assumption of low aspect ratio flows we employ the two-layer shallow water equations wherein the density of the upper layer is specified by a general equation of state involving both salinity and temperature. In all cases we consider fixed volume releases of lighter fluids into heavier ambient fluids whose properties remain unchanged. The complex dynamics of these two-layer systems is investigated using distinguished-limit models combined with a variety of analytical methods, as well as, numerical schemes.