Consider a principal $G$-bundle over a manifold $X$, and consider yet another manifold $B$ endowed with a $G$-action. Everything is assumed to be smooth. The associated bundle $$ \mathcal{B}=P\times_G B $$ is a locally trivial bundle over $X$. My question concerns the space of sections of this associated bundle, i.e., the infinite-dimensional manifold $\mathcal{S} =\Gamma(X,\mathcal{B})$. Assume we have fixed such a section $\phi\in \Gamma(X,\mathcal{B})$. I want to characterize, if possible, the tangent space $$ T_\phi \Gamma(X,\mathcal{B}) $$ of infinitesimal deformations $\dot{\phi}$ of that section, and I want to do it in such a way that $G$-invariant geometric objects on $B$ give corresponding objects in this infinite-dimensional manifold $\Gamma(X,\mathcal{B})$ (will be detailed below).
I want to analyze this tangent space under the light of the following fact: sections of the associated bundle correspond to $G$-equivariant mappings $\tilde{\phi}:P\rightarrow B$.
Preliminary remarks
A section $\phi$ has to always be vertical with respect to the base space. This means that if $\phi_t$ is a parametrized curve of sections of $\mathcal{B}$, $\phi_0=\phi$, then $\frac{d}{dt}|_{t=0} \phi_t$ has to send elements of the base $x\in X$ to vertical vectors in $\mathcal{B}$ attached at $\phi(x)$. We could justify $$ T_\phi \mathcal{S} \simeq \Gamma(X,\phi^*V\mathcal{B}) $$ where $\phi^*V\mathcal{B}$ is the pullback to $X$ under $\phi$ of the vertical bundle of $\mathcal{B}$.
What I am missing
I want a mechanism of lifting $G$-invariant objects on $B$ to objects on $\mathcal{B}$. Specifically, assume we have a $G$-invariant 1-form $\sigma\in \Gamma(B,T^*B)$. Is there a way in which I could build a field of 1-forms over $\mathcal{S}$ by equivariance?
My end goal would to assume $B$ has a Kähler structure $\omega$ and to use this to endow $\mathcal{S}$ some formal Kählerian structure (I'd need to assume $X$ compact with a prescribed volume form) and to make sense of candidate Kähler form on $\mathcal{S}$ via something like $$ \Omega(\dot{\Phi}_1,\dot{\Phi}_2) \sim \int_X \omega(\dot{\phi}_1,\dot{\phi}_2) $$ by either identifying tangent elements $\dot{\Phi}$ in terms of $B$-valued maps or by defining a fiberwise Kähler-form $\omega$ on each fiber of the associated bundle $\mathcal{B}$.