I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able to find such a presentation online. Is it known? And, if so, is it 'basic/common' knowledge in the field?

For example, I think it is possible to get to that description using the degeneracy sets of sections, which seems to be a pretty standard thing.

EDIT

I am only working with flag varieties flags of linear subspaces of $\mathbb{C}^n$ ($U(n)/U(n_1)\times\cdots\times U(n_k)$).