  We apply our earlier work on the higher-dimensional analogue of the Mumford
conjecture to two questions. Inspired by work of Ebert we prove non-triviality
of certain characteristic classes of bundles of smooth closed manifolds.
Inspired by work of Church-Farb-Thibault and Church-Crossley-Giansiracusa we
investigate the dependence of characteristic classes of bundles on
characteristic numbers of its fibre, total space and base space.
