A gauge transformation can be viewed as a section of $P \times_G G$, where $G$ acts via conjugation. Hence the Lie algebra $\mathfrak{gau}$ of infinitesimal gauge transformation is the space of sections of the adjoint bundle $P \times_G \mathfrak{g}$. Via integration the dual space $\mathfrak{gau}^*$ can be identified with the space of sections of the dual bundle; the coadjoint bundle $P \times_G \mathfrak{g}^*$. As usual, an $Ad$-invariant scalar product on $\mathfrak{g}$ yields an isomorphism between the adjoint and the coadjoint bundle. Hence $\mathfrak{gau}$ is self-dual and under this identification the coadjoint action of the gauge group corresponds to the adjoint action. In other words, you just lift the isomorphism $(\mathfrak{g}^*, Ad^*_G) = (\mathfrak{g}, Ad_G)$ to the corresponding bundles and then to the section spaces.

Now equivariance is rather obvious: the curvature (as a $2$-form with values in $Ad\,P$) transforms via the adjoint action under gauge transformations.