  In this note, we give Gysin formulas for partial flag bundles for the
classical groups. We then give Gysin formulas for Schubert varieties in
Grassmann bundles, including isotropic ones. All these formulas are proved in a
rather uniform way by using the step-by-step construction of flag bundles and
the Gysin formula for a projective bundle. In this way we obtain a
comprehensive list of new universal formulas.
