Proper homogeneous $G$-spaces (where $G$ is semisimple algebraic group) over positive characteristic fields $k$ can be divided into two classes, the first one being the flag varieties $G/P$ and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the $D$-affinity of Beilinson and Bernstein fails for varieties of unseparated flags.