We report that phase separation of a fluid mixture, whose temperature is set below the phase separation temperature only locally near one side wall of the container, is strongly delocalized over the entire system if there is density mismatch between the components. We demonstrate that this delocalization is caused by i) gravity-driven convective flow and ii) asymmetric kinetics between fast hydrodynamic coarsening during demixing and slow diffusional mixing through the phase boundary. The relevant physical parameters determining the efficiency of phase separation are clarified.