  We demonstrate how pipe dreams can be applied to the theory of poset
polytopes to produce toric degenerations of flag varieties. Specifically, we
present such constructions for marked chain-order polytopes of Dynkin types A
and C. These toric degenerations also give rise to further algebraic and
geometric objects such as PBW-monomial bases and Newton--Okounkov bodies. We
discuss a construction of the former in the type A case and of the latter in
type C.
