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 .