Let $\pi:P \to M$ be a smooth principal right $G$-bundle, where $G$ is a Lie group. Let $F$ and $F'$ be smooth left $G$-manifolds, and let $a:F \to F'$ be a smooth $G$-equivariant map, meaning that
for every $g \in G$ and every $f \in F$. Define the associated bundles
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\begin{align*}
P \times_G F := (P \times F)/{\sim}
\end{align*}
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and
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\begin{align*}
P \times_G F' := (P \times F')/{\sim'}
\end{align*}
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where $(p,f)\sim(pg,g^{-1}\cdot f)$ and $(p,f')\sim'(pg,g^{-1}\cdot f')$ for all $p \in P$, $g \in G$, $f \in F$, and $f' \in F'$. Then the assignment
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\begin{align*}
A:P \times_G F \to P \times_G F'
\end{align*}
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given by
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\begin{align*}
A([p,f]) := [p,a(f)]
\end{align*}
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is a well-defined smooth bundle map over $M$.
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Moreover, this construction is functorial in the fibre: the identity map $\operatorname{id}_F:F \to F$ induces $\operatorname{id}_{P\times_G F}$, and if $b:F' \to F''$ is another smooth $G$-equivariant map to a smooth left $G$-manifold $F''$, then the map induced by $b \circ a$ is the composition of the maps induced by $a$ and $b$.