  We use the theory of Mori dream spaces to prove that the global Okounkov body
of a Bott-Samelson variety with respect to a natural flag of subvarieties is
rational polyhedral. In fact, we prove more generally that this holds for any
Mori dream space which admits a flag of Mori dream spaces satisfying a certain
regularity condition. As a corollary, Okounkov bodies of effective line bundles
over Schubert varieties are shown to be rational polyhedral. In particular, it
follows that the global Okounkov body of a flag variety $G/B$ is rational
polyhedral.
  As an application we show that the asymptotic behaviour of dimensions of
weight spaces in section spaces of line bundles is given by the counting of
lattice points in polytopes.
