We investigate the influence of buoyancy on electroconvection at an ion-exchange membrane in an aqueous electrolyte solution. Electrokinetic instabilities (EKIs) and Rayleigh-Bénard (RB) convection are both known to mix the appearing concentration gradient layer and overcome the limiting current arising from diffusional limitations. The different physics, as well as the interplay between them, are investigated by electrical, flow, and concentration characterization. In the buoyancy stable orientation, an EKI mixing layer, having a low concentration, grows till saturated size. In the buoyancy unstable orientation, RB occurs and dominates the advective transport due to the large system size. When current density i<5ilim, RB mixes the system and EKI does not arise. If i>5ilim EKI starts before RB and hastens the onset of RB. Upon onset of RB, EKI is suppressed while the overall resistance is still decreased. The onset times of EKI and RB could be predicted using a simple diffusion-migration model based on Fick's second law.