  A Newton-Okounkov body is a convex body constructed from a projective variety
with a globally generated line bundle and with a higher rank valuation on the
function field, which gives a systematic method of constructing toric
degenerations of projective varieties. Its combinatorial properties heavily
depend on the choice of a valuation, and it is a fundamental problem to relate
Newton-Okounkov bodies associated with different kinds of valuations. In this
paper, we address this problem for flag varieties using the framework of
combinatorial mutations which was introduced in the context of mirror symmetry
for Fano manifolds. By applying iterated combinatorial mutations, we connect
specific Newton-Okounkov bodies of flag varieties including string polytopes,
Nakashima-Zelevinsky polytopes, and FFLV polytopes.
