In recent years, the theme of asymptotically flat spacetimes has come back to the fore, fueled by the discovery of gravitational waves and the growing interest in what flat-space holography could be. In this quest, the original tools pertaining to the standard anti-de Sitter conformal field theory duality have been insufficiently exploited. After outlining a synoptic survey of Carrollian geometric structures, I will show how Ricci-flat spacetimes are generally reached as a limit of Einstein geometries and how they are in fact constructed by means of data defined on the conformal Carrollian boundary that is null infinity. These data, infinite in number, are obtained as the coefficients of the Laurent expansion of the energy-momentum tensor in powers of the cosmological constant. Such an approach puts this tensor back at the heart of the asymptotically flat dynamics, clearing the way for its subsequent flat-space holographic use.