  Nominal sets provide a framework to study key notions of syntax and semantics
such as fresh names, variable binding and $\alpha$-equivalence on a
conveniently abstract categorical level. Coalgebras for endofunctors on nominal
sets model, e.g., various forms of automata with names as well as infinite
terms with variable binding operators (such as $\lambda$-abstraction). Here, we
first study the behaviour of orbit-finite coalgebras for functors $\bar F$ on
nominal sets that lift some finitary set functor $F$. We provide sufficient
conditions under which the rational fixpoint of $\bar F$, i.e. the collection
of all behaviours of orbit-finite $\bar F$-coalgebras, is the lifting of the
rational fixpoint of $F$. Second, we describe the rational fixpoint of the
quotient functors: we introduce the notion of a sub-strength of an endofunctor
on nominal sets, and we prove that for a functor $G$ with a sub-strength the
rational fixpoint of each quotient of $G$ is a canonical quotient of the
rational fixpoint of $G$. As applications, we obtain a concrete description of
the rational fixpoint for functors arising from so-called binding signatures
with exponentiation, such as those arising in coalgebraic models of infinitary
$\lambda$-terms and various flavours of automata.
