  Given a smooth Mori dream space $X$ we construct a model dominating all the
small $\mathbb{Q}$-factorial modifications via tropicalization. This
construction allows us to recover a Minkowski basis for the Newton-Okounkov
bodies of divisors on $X$ and hence the movable cone of $X$. The existence of
such basis allows us to prove the polyhedrality of the global Newton-Okounkov
body and the existence of toric degenerations.
