  For a connected abelian Lie group T acting on a Poisson manifold (Y,{\pi}) by
Poisson isomorphisms, the T-leaves of {\pi} in Y are, by definition, the orbits
of the symplectic leaves of {\pi} under T, and the leaf stabilizer of a T-leaf
is the subspace of the Lie algebra of T that is everywhere tangent to all the
symplectic leaves in the T-leaf. In this paper, we first develop a general
theory on T-leaves and leaf stabilizers for a class of Poisson structures
defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply
the general theory to four series of holomorphic Poisson structures on products
of flag varieties and related spaces of a complex semi-simple Lie group G. We
describe their T-leaf decompositions, where T is a maximal torus of G, in terms
of (open) extended Richardson varieties and extended double Bruhat cells
associated to conjugacy classes of G, and we compute their leaf stabilizers and
the dimension of the symplectic leaves in each T -leaf.
