We develop and analyze in this paper a momentum convection operator for variable density flows, and apply it to obtain a finite volume scheme for the Euler equations. The mesh is composed of triangular and quadrangular cells, in the two-dimensional case, and of hexahedral, tetrahedral, prismatic and pyramidal cells in three space dimensions. The approximation is staggered: the scalar variables (pressure, density and internal energy) are associated with the cells while the velocity approximation is face-centred. The derivation of the momentum convection operator extends to pyramids and prisms an already proposed construction for the other above-mentioned cells. The resulting operator takes the form of a finite volume operator, but is obtained by an algebraic process using as input the mass fluxes through the primal faces appearing in the mass balance for the definition of the velocity fluxes, with the only guideline to satisfy a discrete local kinetic energy identity. Its consistency thus deserves to be studied, and we show that this process yields a consistent convection operator in the Lax-Wendroff sense. Numerical tests confirm the expected scheme convergence, with a first-order rate on a pure shock solution.