  Gravity in $4d$ asymptotically flat spacetime constitutes the archetypal
example of a gravitational system with leaky boundary conditions. Pursuing our
analysis of [1], we argue that the holographic description of such a system
requires the coupling of the dual theory living at null infinity to some
external sources encoding the radiation reaching the conformal boundary and
responsible for the non-conservation of the charges. In particular, we show
that the sourced Ward identities of a conformal Carrollian field theory living
at null infinity reproduce the BMS flux-balance laws. We also derive the
general form of low-point correlation functions for conformal Carrollian field
theories and exhibit a new branch of solutions, which is argued to be the
relevant one for holographic purposes. We then relate our Carrollian approach
to the celestial holography proposal by mapping the Carrollian Ward identities
to those constraining celestial operators through a suitable integral
transform.
