  The analytical description of the dynamics in models with discrete variables
(e.g. Ising spins) is a notoriously difficult problem, that can be tackled only
under some approximation. Recently a novel variational approach to solve the
stationary dynamical regime has been introduced by Pelizzola [Eur. Phys. J. B,
86 (2013) 120], where simple closed equations are derived under mean-field
approximations based on the cluster variational method. Here we propose to use
the same approximation based on the cluster variational method also for the
non-stationary regime, which has not been considered up to now within this
framework. We check the validity of this approximation in describing the
non-stationary dynamical regime of several Ising models defined on
Erdos-R\'enyi random graphs: we study ferromagnetic models with symmetric and
partially asymmetric couplings, models with random fields and also spin glass
models. A comparison with the actual Glauber dynamics, solved numerically,
shows that one of the two studied approximations (the so-called 'diamond'
approximation) provides very accurate results in all the systems studied. Only
for the spin glass models we find some small discrepancies in the very low
temperature phase, probably due to the existence of a large number of
metastable states. Given the simplicity of the equations to be solved, we
believe the diamond approximation should be considered as the 'minimal
standard' in the description of the non-stationary regime of Ising-like models:
any new method pretending to provide a better approximate description to the
dynamics of Ising-like models should perform at least as good as the diamond
approximation.
