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In types C and B we construct new families of toric degenerations and Newton--Okounkov bodies of flag varieties and also of PBW-monomial bases in irreducible representations. The constructed objects are given by certain poset polytopes which in both types form families interpolating between the respective Gelfand--Tsetlin and FFLV polytopes. In type C these are the MCOPs of Fang and Fourier, in type B we introduce a new family of poset polytopes. Our main definitions and proofs rely on a combinatorial connection between poset polytopes and pipe dreams.
Makedonskyi et al. (Sun,) studied this question.
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