  The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber
$M$ depend only on the underlying $\tau_M$-fibration, meaning the family of
vector bundles formed by the tangent bundles of the fibers. This motivates a
closer study of the classifying space for $\tau_M$-fibrations, $Baut(\tau_M)$,
and its cohomology ring, i.e., the ring of characteristic classes of
$\tau_M$-fibrations. For a bundle $\xi$ over a simply connected Poincar\'e
duality space, we construct a relative Sullivan model for the universal
orientable $\xi$-fibration together with explicit cocycle representatives for
the characteristic classes of the canonical bundle over its total space. This
yields tools for computing the rational cohomology ring of $Baut(\xi)$ as well
as the subring generated by the generalized Miller-Morita-Mumford classes. To
illustrate, we carry out sample computations for spheres and complex projective
spaces. We discuss applications to tautological rings of simply connected
manifolds and to the problem of deciding whether a given $\tau_M$-fibration
comes from a manifold bundle.
