We consider the demixing of a binary fluid mixture, under gravity, which is steadily driven into a two-phase region by slowly ramping the temperature. We assume, as a first approximation, that the system remains spatially isothermal, and we examine the interplay of two competing nonlinearities. One of these arises because the supersaturation is greatest far from the meniscus, creating inversions of the density which can lead to fluid motion; although isothermal, this is somewhat like the Bénard problem (a single-phase fluid heated from below). The other is the intrinsic diffusive instability which results either in nucleation or in spinodal decomposition at large supersaturations. Experimental results on a simple binary mixture show interesting oscillations in heat capacity and optical properties, for a wide range of ramp parameters. We argue that these oscillations arise under conditions where both nonlinearities are important.