  The product monomial crystal was defined by Kamnitzer, Tingley, Webster,
Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$,
and depends on a collection of parameters $\mathbf{R}$. We show that a family
of truncations of this crystal are Demazure crystals, and give a Demazure-type
formula for the character of each truncation, and the crystal itself. This
character formula shows that the product monomial crystal is the crystal of a
generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar.
In type $A$, we show the product monomial crystal is the crystal of a
generalised Schur module associated to a column-convex diagram depending on
$\mathbf{R}$.
