  In previous work we have shown the existence of a dynamical horizon or
marginally trapped tube (MOTT) containing a given strictly stable marginally
outer trapped surface (MOTS). In this paper we show some results on the global
behavior of MOTTs assuming the null energy condition. In particular we show
that MOTSs persist in the sense that every Cauchy surface in the future of a
given Cauchy surface containing a MOTS also must contain a MOTS. We describe a
situation where the evolving outermost MOTS must jump during the coalescence of
two seperate MOTSs. We furthermore characterize the behavior of MOTSs in the
case that the principal eigenvalue vanishes under a genericity assumption. This
leads to a regularity result for the tube of outermost MOTSs under the
genericity assumption. This tube is then smooth up to finitely many jump times.
Finally we discuss the relation of MOTSs to singularities of a space-time.
