Electromagnetic (EM) heat exchangers (HX) are critical components in power beaming applications where EM waves are radiated towards an EM HX, which then converts incident energy into heat or mechanical work. An EM HX consists of a lossy ceramic undergoing EM heating, and a fluid flow maintaining thermal contact transfers heat from the ceramic. Temperatures during high-power EM processing of ceramics materials such as zirconia suggest that liquids would be in gaseous phase, so models of EM HX with compressible gas dynamics may provide insights into experimental scenario. As a first step, we consider an EM HX such that plane Poiseuille flow of an incompressible coolant whose density drops linearly with temperature is situated above a lossy ceramic material. Compressible effects are negligible, but density gradients within the fluid can give rise to buoyancy-driven flow under the action of gravity, which may affect the performance of the device. We determine the power of incident waves at which Bénard convection is initiated, through a linear stability analysis, in the fluid layer. We show that in case of temperature dependent ceramic loss factor, perturbations in temperature give rise to an electric (fringe) field which then feeds back into the system promoting the Bénard instability.