  We study the relationship between general dynamical Poisson groupoids and Lie
quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with
a reductive decomposition, we extend the description of the moduli space of
classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each
gauge orbit, an explicit analytic representative l. We translate the notion of
duality for dynamical Poisson groupoids into a duality for Lie
quasi-bialgebras. It is shown that duality maps the dynamical Poisson groupoid
for l and a Lie quasi-bialgebra to the dynamical Poisson groupoid for the dual
data.
