  We study the non-equilibrium cumulants of the chiral order parameter field
({\sigma} field) in different phase transition scenarios via Langevin dynamics.
Cumulants up to fourth-order have been calculated based on the
spacetimedependent {\sigma} configurations from the event-by-event numerical
simulations. By limiting the cooling of the system in a Hubble-like way, the
out-of-equilibrium cumulants illustrate clear memory effects during the
evolution. Both the signs and the magnitudes of the high-order cumulants differ
from the equilibrium ones below the phase transition temperature. Especially,
the dynamical cumulants grow more intensively from the first-order phase
transition side than they do from the crossover side. In addition, analysis of
the high-order offequilibrium cumulants on the hypothetical freeze-out lines
present non-monotonic curves in the large chemical potential region.
