The theory of polyptych lattices is a framework to obtain a family of toric degenerations whose polytopes are related by piecewise-linear transformations. It can be regarded as a generalization of toric degenerations arising from cluster algebras. In this paper, we study polyptych lattices consisting of transfer maps for marked chain-order polytopes, and obtain a family of toric degenerations of a projective variety to marked chain-order polytopes for the Gelfand-Tsetlin poset. We also compute the Cox ring of this projective variety.