A convective vertical mixing scheme rooted in the Eddy-Diffusivity Mass-Flux (EDMF) approach is carefully derived from first principles. This type of closure involves separating vertical turbulent fluxes into two components: an eddy-diffusivity (ED) term that addresses local small-scale mixing in a near isotropic environment, and a mass-flux (MF) transport term that accounts for the non-local transport performed by vertically coherent plumes within the environment. Using the multi-fluid averaging underlying the MF concept, we present consistent energy budgets between resolved and subgrid scales for seawater and dry atmosphere. We show that when using an EDMF scheme, closed energy budgets can be recovered if: (i) bulk production terms of turbulent kinetic energy (TKE) by shear, buoyancy and transport include MF contributions; (ii) boundary conditions are consistent with EDMF, to avoid spurious energy fluxes at the boundary. The performance of the energetically consistent EDMF scheme is evaluated against Large Eddy Simulations (LES) and observational data of oceanic convection. Notably, energetic consistency is key to obtaining accurate TKE and turbulent transport of TKE profiles when compared to LES data. Throughout the theoretical development of the scheme, we maintain transparency regarding underlying assumptions and systematically assess their validity in the light of LES data.