The Atlantic meridional overturning circulation (MOC) is traditionally monitored in terms of zonally integrated transport either in depth space or in density space. While this view has the advantage of simplicity, it obscures the rich and complex three-dimensional structure, so that the exact physics of the downwelling and upwelling branch remains poorly understood. The near-equivalence of the depth- and density-space MOC in the subtropics suggests that vertical and diapycnal volumes transports are intimately coupled, whereas the divergence of these two metrics at higher latitudes indicates that any such coupling is neither instantaneous nor local. Previous work has characterized the surface buoyancy forcing and mixing processes which drive diapycnal volume transport. Here, we develop a new analytical decomposition of vertical volume transport based on the vorticity budget. We show that most terms can be estimated from observations and provide additional insights from a high-resolution numerical simulation of the North Atlantic. Our analysis highlights the roles of 1) relative vorticity advection for the sinking of overflow water at the northern subpolar North Atlantic boundaries and 2) the geostrophic β effect for the sinking of dense waters in the intergyre region. These results provide insights into the coupling between density- and depth-space overturning circulations.