Naturality of minimal model of a fibre bundle $\require{AMScd}$
For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's
$$  
\begin{CD}
        A_{PL}(B) @>>> A_{PL}(E) @>>> A_{PL}(F) \\
        @AAA @AAA @AAA \\
        (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F})
\end{CD}
$$
where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c
Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.
 A: I don't know what you are looking for exactly. There's Proposition 15.8 in the book Rational Homotopy Theory by Félix, Halperin, and Thomas (and I don't doubt that there is an analogous proposition for non simply connected spaces in the sequel but I haven't checked). There is a little bit of setup to do so please bear with me.
Suppose that you have two fibrations and a morphism of fiber sequences:
$$\require{AMScd}
\begin{CD}
F @>>> E @>p>> B \\
@VhVV @VgVV @VfVV \\
F' @>>> E' @>p'>> B'
\end{CD}$$
where for example you have base points $x_0 \in B$, $x_0' \in B'$ such that $f(x_0) = x_0'$, $F = p^{-1}(x_0)$ and $F' = (p')^{-1}(x_0')$.
Suppose that you are given a model for the fibration
$$\begin{CD}
(\Lambda W, \bar{d}) @<<< (\Lambda V \otimes \Lambda W, D) @<<< (\Lambda V, d) \\
@V{\bar{m}}VV @VMVV @VmVV \\
A_{PL}(F) @<<< A_{PL}(E) @<{p^*}<< A_{PL}(B)
\end{CD}$$
and the same thing with primes everywhere. Suppose also that you have a model $\psi : (\Lambda V', d') \to (\Lambda V, d)$ for $f$, i.e. $m\psi = f^* m'$.
Then you get a map from the pushout:
$$\xi : (\Lambda V, d) \otimes_{(\Lambda V', d')} (\Lambda V' \otimes \Lambda W', D) \to A_{PL}(E).$$
You can rewrite the domain of this pushout as $(\Lambda V \otimes \Lambda W', \delta)$ for a differential $\delta$. You moreover have a commutative diagram:
$$\begin{CD}
(\Lambda V, d) @>>> (\Lambda V \otimes \Lambda W', \delta) @>>> (\Lambda W', \bar{d'} \\
@VVV @VVV @VVV \\
A_{PL}(B') @>>> A_{PL}(E) @>>> A_{PL}(F)
\end{CD}$$
The proposition is that if $h^* : H^*(F') \to H^*(F)$ is an isomorphism (with rational coefficients), then this last diagram is a model for the fibration $F \to E \to B'$.
Anyway the upshot is that if you have a model for a fibration and you pull back the fibration along some map of which you also have a model, then you get a model for the pullback. (And I certainly hope that I put all the primes where they belong.)
