  Kohnert polynomials are polynomials indexed by unit cell diagrams in the
first quadrant defined earlier by the author and Searles that give a common
generalization of Schubert polynomials and Demazure characters for the general
linear group. Demazure crystals are certain truncations of normal crystals
whose characters are Demazure characters. For each diagram satisfying a
southwest condition, we construct a Demazure crystal whose character is the
Kohnert polynomial for the given diagram, resolving an earlier conjecture of
the author and Searles that these polynomials expand nonnegatively into
Demazure characters. We give explicit formulas for the expansions with
applications including a characterization of those diagrams for which the
corresponding Kohnert polynomial is a single Demazure character.
