  We study multiplicative quiver varieties associated to specific extensions of
cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is
characterised by quasi-Hamiltonian algebras related to these quivers, which
were studied by Van den Bergh for an arbitrary quiver. We show that the spaces
are generically isomorphic to the case $m=1$ corresponding to an extended
Jordan quiver. This provides a set of local coordinates, which we use to
interpret integrable systems as spin variants of the trigonometric
Ruijsenaars-Schneider system. This generalises to new spin cases recent works
on classical integrable systems in the Ruijsenaars-Schneider family.
