The universal fibration with fibre $X$ in rational homotopy theory

Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_infty colon UE to mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ omega colon mathrm{aut}_1(mathrm{Baut}_1(X_{mathbb Q})) to mathrm{Baut}_1(X_{mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $mathrm{Baut}_1(X_{mathbb Q})$ as a consequence. We also prove that ${mathbb C} P^n_{mathbb Q}$ cannot be realized as $mathrm{Baut}_1(X_{mathbb Q})$ for $n leq 4$ and $X$ with finite-dimensional rational homotopy groups.