Diagonals of flag bundles

We express the diagonals of projective, Grassmann and, more generally, flag bundles of type (A) using the zero schemes of some vector bundle sections, and do the same for their single point subschemes. We discuss diagonal and point properties of these flag bundles. We study when the complex manifolds G/B for other groups have the point and diagonal properties. We discuss explicit formulas for the classes of diagonals of the varieties G/B.