  We show that we can derive the asymptotic Einstein's equations that arises at
order $1/r$ in asymptotically flat gravity purely from symmetry considerations.
This is achieved by studying the transformation properties of functionals of
the metric and the stress-energy tensor under the action of the Weyl BMS group,
a recently introduced asymptotic symmetry group that includes arbitrary
diffeomorphisms and local conformal transformations of the metric on the
2-sphere. Our derivation, which encompasses the inclusion of matter sources,
leads to the identification of covariant observables that provide a definition
of conserved charges parametrizing the non-radiative corner phase space. These
observables, related to the Weyl scalars, reveal a duality symmetry and a
spin-2 generator which allow us to recast the asymptotic evolution equations in
a simple and elegant form as conservation equations for a null fluid living at
null infinity. Finally we identify non-linear gravitational impulse waves that
describe transitions among gravitational vacua and are non-perturbative
solutions of the asymptotic Einstein's equations. This provides a new picture
of quantization of the asymptotic phase space, where gravitational vacua are
representations of the asymptotic symmetry group and impulsive waves are
encoded in their couplings.
