  We prove that on a Bott-Samelson variety $X$ every movable divisor is nef.
This enables us to consider Zariski decompositions of effective divisors, which
in turn yields a description of the Mori chamber decomposition of the effective
cone. This amounts to information on all possible birational morphisms from
$X$. Applying this result, we prove the rational polyhedrality of the global
Newton-Okounkov body of a Bott-Samelson variety with respect to the so called
`horizontal' flag. In fact, we prove the stronger property of the finite
generation of the corresponding global value semigroup.

Algebraic Geometry