  One definition of key polynomials is as the weight generating functions of
key tableaux. Assaf and Schilling introduced a crystal structure on key
tableaux and related it to Morse--Schilling crystals on reduced factorizations
for permutations via weak Edelman--Greene insertion. In this paper, we consider
generalizations of both crystals depending on a flag. We extend weak EG
insertion to a bijection between our flagged objects and show that the
recording tableau gives a crystal isomorphism. As an application, we show that
flagged key tableaux have a natural Demazure crystal structure, whose
characters recover Reiner and Shimozono's flagged key polynomials.
