  For noncompact semisimple Lie groups $G$ we study the dynamics of the actions
of their discrete subgroups $\Gamma<G$ on the associated partial flag manifolds
$G/P$. Our study is based on the observation that they exhibit also in higher
rank a certain form of convergence type dynamics. We identify geometrically
domains of proper discontinuity in all partial flag manifolds. Under certain
dynamical assumptions equivalent to the Anosov subgroup condition, we establish
the cocompactness of the $\Gamma$-action on various domains of proper
discontinuity, in particular on domains in the full flag manifold $G/B$. We
show in the regular case (of $B$-Anosov subgroups) that the latter domains are
always nonempty if if $G$ has (locally) at least one noncompact simple factor
not of the type $A_1, B_2$ or $G_2$.
