  We reconsider formulating $D$ dimensional gauge theories, with the focus on
the case of gravity theories, in spacetimes with boundaries. We extend
covariant phase space formalism to the cases in which boundaries are allowed to
fluctuate. We analyze the symplectic form, the freedoms (ambiguities), and its
conservation for this case. We show that boundary fluctuations render all the
surface charges integrable. We study the algebra of charges and its central
extensions, charge conservation, and fluxes. We briefly comment on memory
effects and questions regarding semiclassical aspects of black holes in the
fluctuating boundary setup.
