  We extend the notion of Gibbsianness for mean-field systems to the set-up of
general (possibly continuous) local state spaces. We investigate the Gibbs
properties of systems arising from an initial mean-field Gibbs measure by
application of given local transition kernels. This generalizes previous
case-studies made for spins taking finitely many values to the first step in
direction to a general theory, containing the following parts: (1) A formula
for the limiting conditional probability distributions of the transformed
system. It holds both in the Gibbs and non-Gibbs regime and invokes a
minimization problem for a "constrained rate-function". (2) A criterion for
Gibbsianness of the transformed system for initial Lipschitz-Hamiltonians
involving concentration properties of the transition kernels. (3) A continuity
estimate for the single-site conditional distributions of the transformed
system. While (2) and (3) have provable lattice-counterparts, the
characterization of (1) is stronger in mean-field. As applications we show
short-time Gibbsianness of rotator mean-field models on the (q-1)-dimensional
sphere under diffusive time-evolution and the preservation of Gibbsianness
under local coarse-graining of the initial local spin space.
