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Abstract Asymptotically flat spacetimes have been studied in five separate regions: future/past timelike infinity $i^{\pm }$, future/past null infinity, and spatial infinityi⁰. We formulate assumptions and definitions such that the five infinities share a single Bondi–Metzner–Sachs (BMS) group of asymptotic symmetries and associated charges. We show how individual ingoing/outgoing massive bodies may be ascribed initial/final BMS charges and derive global conservation laws stating that the change in total charge is balanced by the corresponding radiative flux. This framework provides a foundation for the study of asymptotically flat spacetimes containing ingoing and outgoing massive bodies, i.e. for generalized gravitational scattering. Among the new implications are rigorous definitions for quantities like initial/final spin, scattering angle, and impact parameter in multi-body spacetimes, without the use of any preferred background structure.