  Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag
variety. The projections of Richardson varieties from the full flag variety
form a stratification of $G/P$. We show that the closure partial order of
projected Richardson varieties agrees with that of a subset of Schubert
varieties in the affine flag variety of $G$. Furthermore, we compare the
torus-equivariant cohomology and $K$-theory classes of these two
stratifications by pushing or pulling these classes to the affine Grassmannian.
Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian
of type $A$.
