  This paper is devoted to a study of the relative version of a Mori dream
space (MDS for short), which was first introduced by Andreatta and
Wi\'{s}newski and will be called Mori dream morphism (MDM) in this paper. An
MDM is defined to be an algebraic fiber space between normal quasi-projective
varieties $ X \to U$ such that
$\operatorname{Pic}(X/U)_{\mathbb{Q}}=\operatorname{N} ^1(X/U)_{\mathbb{Q}}$
and the (relative) movable cone is decomposed into the semi-ample cones of
finitely many small $\mathbb{Q}$-factorial modifications, each of which is
assumed to be rational polyhedral. An MDS is an MDM where $U$ is a point. We
prove that the relative $D$-MMP runs and terminates in either a good
$D$-minimal model or a $D$-Mori fiber space for an arbitrary divisor $D$ on an
MDM, and that an algebraic fiber space satisfying
$\operatorname{Pic}(X/U)_{\mathbb{Q}}= \operatorname{N} ^1(X/U)_{\mathbb{Q}}$
is an MDM if and only if a Cox sheaf is finitely generated over $U$. These are
generalizations to an MDM of the fundamental results by Hu and Keel for an MDS.
We also show that if the composition of two algebraic fiber spaces $f$ and $g$
is an MDM, then so are $f$ and $g$. In the end we investigate base changes of
MDMs. We prove that a base change of an MDM by a proper flat morphism is again
an MDM, provided that the base change is normal $\mathbb{Q}$-factorial and that
any $\mathbb{Q}$-line bundle on the base change descends to the original MDM.
