  We consider positive operator semigroups on ordered Banach spac\-es and study
the relation of their long time behaviour to two different domination
properties. First, we analyse under which conditions almost periodicity and
mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups
which are asymptotically dominated by $\mathcal{T}$. Then, we consider
semigroups whose orbits asymptotically dominate a positive vector and show that
this assumption is often sufficient to conclude strong convergence of the
semigroup as time tends to infinity.
  Our theorems are applicable to time-discrete as well as time-continuous
semigroups. They generalise several results from the literature to considerably
larger classes of ordered Banach spaces.
