Let G be a classical complex Lie group, P any parabolic subgroup of G, and X=G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a vector space. In the mid 1990s, Fulton asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. We will explain our recent combinatorially explicit solution to this question.