  The dressing field method is a tool to reduce gauge symmetries. Here we
extend it to cover the case of diffeomorphisms. The resulting framework is a
systematic scheme to produce Diff(M)-invariant objects, which has a natural
relational interpretation.
  Its precise formulation relies on a clear understanding of the bundle
geometry of field space. By detailing it, among other things we stress the
geometric nature of field-independent and field-dependent diffeomorphisms, and
highlight that the heuristic "extended bracket" for field-dependent vector
fields often featuring in the covariant phase space literature can be
understood as arising from the Fr\"olicher-Nijenhuis bracket. Furthermore, by
articulating this bundle geometry with the covariant phase space approach, we
give a streamlined account of the elementary objects of the (pre)symplectic
structure of a Diff(M)-theory: Noether charges and their bracket, as induced by
the standard prescription for the presymplectic potential and 2-form. We give
conceptually transparent expressions allowing to read the integrability
conditions and the circumstances under which the bracket of charge is Lie, and
the resulting Poisson algebras of charges are central extensions of the Lie
algebras of field-independent ($\mathfrak{diff}(M)$) and field-dependent vector
fields.
  We show that, applying the dressing field method, one obtains a
Diff(M)-invariant and manifestly relational formulation of a general
relativistic field theory. Relying on results just mentioned, we easily derive
the "dressed" (relational) presymplectic structure of the theory. This
reproduces or extends results from the gravitational edge mode and
gravitational dressing literature. In addition to simplified technical
derivations, the conceptual clarity of the framework supplies several insights
and allows us to dispel misconceptions.
