Defining an energy-momentum tensor for asymptotically flat spacetimes seems to be of interest in its own right, or for possible holographic applications. That said, the role it could play is less clear than for anti de Sitter, bearing in mind that in the flat case an infinite number of data are needed on scri to reconstruct a solution. I will show that besides the Carrollian geometric structure at null infinity, this infinite set dubbed Carrollian momenta emerge as the Laurent coefficients of the AdS energy-momentum tensor expanded in powers of the cosmological constant. This analysis highlights the role of the boundary Cotton tensor and its involvement in duality properties such as electric-magnetic, Ehlers, etc.