  This article establishes alcove walk models for intersections of Schubert
varieties and partially semi-infinite orbits in the affine Grassmannian of a
split reductive group (we call such intersections parabolic Mirkovi\'c-Vilonen
intersections). More precisely, we describe explicit cellular pavings of these
intersections, indexed by certain positively-folded alcove walks. We prove a
parametrization of the irreducible components of maximal possible dimension, in
terms of alcove walks of maximal possible dimension. We then deduce a new
combinatorial description of branching to Levi subgroups of irreducible highest
weight representations, and in particular we give a new algorithm for computing
the characters of such representations.
