  A simple and very flexible variational approach to the out-of-equilibrium
quantum dynamics in strongly correlated electron systems is introduced through
a time-dependent Gutzwiller wavefunction. As an application, we study the
simple case of a sudden change of the interaction in the fermionic Hubbard
model and find at the mean field level an extremely rich behaviour. In
particular, a dynamical transition between small and large quantum quench
regimes is found to occur at half-filling, in accordance with the analysis of
Eckstein {\sl et al.}, Phys. Rev. Lett. {\bf 103}, 056403 (2009), obtained by
dynamical mean field theory, that turns into a crossover at any finite doping.
