  We derive a coupled set of equations that describe the non-equilibrium
evolution of cumulants of critical fluctuations for space-time trajectories on
the cross-over side of the QCD phase diagram. In particular, novel expressions
are obtained for the non-equilibrium evolution of non-Gaussian Skewness and
Kurtosis cumulants. Utilizing a simple model of the space-time evolution of a
heavy- ion collision, we demonstrate that, depending on the relaxation rate of
critical fluctuations, Skewness and Kurtosis can differ significantly in
magnitude as well as in sign from equilibrium expectations. Memory effects are
important and shown to persist even for trajectories that skirt the edge of the
critical regime. We use phenomenologically motivated parameterizations of
freeze-out curves, and of the beam energy dependence of the net baryon chemical
potential, to explore the implications of our model study for the critical
point search in heavy-ion collisions.
