  We propose a framework to study local gauge theories on manifolds with
boundaries and asymptotic symmetries, which is based on representing them as
so-called gauge PDEs. These objects extend the conventional BV-AKSZ
sigma-models to the case of not necessarily topological and diffeomorphism
invariant systems and are known to behave well with respect to restrictions to
submanifolds and boundaries. We introduce the notion of gauge PDE with
boundaries, which takes into account generic boundary conditions, and apply the
framework to asymptotically flat gravity. In so doing we start with a suitable
representation of gravity as a gauge PDE with boundaries which implements the
Penrose's description of asymptotically simple spacetimes. We then derive the
minimal model of the gauge PDE induced on the boundary and observe that it
provides the Cartan (frame-like) description of a (curved) conformal Carollian
structure on the boundary. Furthermore, imposing a suitable version of the
familiar boundary conditions in the induced boundary gauge PDE immediately
leads to the conventional BMS algebra of asymptotic symmetries. Finally, we
briefly sketch the construction in the case of asymptotically (A)dS gravity.
