  We continue our study of Gibbs-non-Gibbs dynamical transitions. In the
present paper we consider a system of Ising spins on a large discrete torus
with a Kac-type interaction subject to an independent spin-flip dynamics
(infinite-temperature Glauber dynamics). We show that, in accordance with the
program outlined in \cite{vEFedHoRe10}, in the thermodynamic limit
Gibbs-non-Gibbs dynamical transitions are \emph{equivalent} to bifurcations in
the set of global minima of the large-deviation rate function for the
trajectories of the empirical density \emph{conditional} on their endpoint.
More precisely, the time-evolved measure is non-Gibbs if and only if this set
is not a singleton for \emph{some} value of the endpoint. A partial description
of the possible scenarios of bifurcation is given, leading to a
characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp
transition times.
  Our analysis provides a conceptual step-up from our earlier work on
Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model, where the
mean-field interaction allowed us to focus on trajectories of the empirical
magnetization rather than the empirical density.
