  We study transport processes on infinite networks. The solution of these
processes can be modeled by an operator semigroup on a suitable Banach space.
Classically, such semigroups are strongly continuous and therefore their
asymptotic behaviour is quite well understood. However, recently new examples
of transport processes emerged where the corresponding semigroup is not
strongly continuous. Due to this lack of strong continuity, there are currently
only few results on the long-term behaviour of these semigroups. In this paper,
we discuss the asymptotic behaviour for a certain class of these transport
processes. In particular, it is proved that the solution semigroups behave
asymptotically periodic with respect to the operator norm as a consequence of a
more general result on the long-term behaviour by positive semigroups
containing a multiplication operator. Furthermore, we revisit known results on
the asymptotic behaviour of transport processes on infinite networks and prove
the asymptotic periodicity of their extensions to the space of bounded
measures.
