The BV-BFV formalism is a combination of the BV approach to quantisation of Lagrangian field theories with local symmetries and the BFV approach to quantisation of constrained Hamiltonian systems. It aims to assign compatible bulk-boundary cohomological data to a Lagrangian field theory on a manifold with boundary (and higher codimension strata), in view of a perturbative quantisation scheme that is compatible with cutting and gluing. General Relativity (GR), seen as a field theory, is a very important example to phrase within this setting, and one in which interesting new insight and complications emerge already at the classical level. In this talk I will present a summary of investigations on GR within the BV-BFV formalism, as well as other diffeomorphism-invariant theories, which have given access to rich and nontrivial information about the boundary structure of gravitational models. I will argue that the featured examples present unexpected insight for the program of quantisation with boundary (and higher strata).
