Definition of Buoyancy Flux

Your confusion likely comes from it's use in equations. Really, this is the buoyancy flux per unit mass. In other words, the term in the governing equation will often appear as $rho B$.

Without the specific paper(s) you are reading however, I cannot be sure of the definition.

Think of the buoyancy of a parcel as minus the difference between its weight and that of the fluid it displaces. You experience the force of gravity as a downward force that determines your weight. A parcel (e.g. a Helium-filled balloon) that is less dense than it's surroundings is positively buoyant - it seems to have negative weight.

So the buoyancy flux through an area is really just the negative of the overall weight-flux. More dense air can be moving downwards, less dense upwards for a positive buoyancy flux. As previously stated, it is usually stated per unit mass, which is the cause of confusion (what's left is 'g', the acceleration due to gravity) and it's slightly confusing to talk about a flux of acceleration. This 'weight-flux' also represents a flux of potential energy, and the work done by the gravitational force in generating it produces kinetic energy; this is a vital term in the turbulent kinetic energy budget.