  In this paper, we show the compatibility of the so-called "dressing field
method", which allows a systematic reduction of gauge symmetries, with the
inclusion of diffeomorphisms in the BRST algebra of a gauge theory. The
robustness of the scheme is illustrated on two examples where Cartan
connections play a significant role. The former is General Relativity, while
the latter concerns the second-order conformal structure where one ends up with
a BRST algebra handling both the Weyl residual symmetry and diffeomorphisms of
spacetime. We thereby provide a geometric counterpart to the BRST cohomological
treatment used in~\cite{Boulanger1} in the construction of a Weyl covariant
tensor calculus.
