  We study the out of equilibrium dynamics of the Fermionic Hubbard Model
induced by a linear ramp of the repulsive interaction $U$ from the metallic
state through the Mott transition. To this extent we use a time dependent
Gutzwiller variational method and complement this analysis with the inclusion
of quantum fluctuations at the leading order, in the framework of a $Z_2$ slave
spin theory. We discuss the dynamics during the ramp and the issue of
adiabaticity through the scaling of the excitation energy with the ramp
duration $\tau$. In addition, we study the dynamics for times scales longer
than the ramp time, when the system is again isolated and the total energy
conserved. We establish the existence of a dynamical phase transition analogous
to the one present in the sudden quench case and discuss its properties as a
function of final interaction and ramp duration. Finally we discuss the role of
quantum fluctuations on the mean field dynamics for both long ramps, where spin
wave theory is sufficient, and for very short ramps, where a self consistent
treatment of quantum fluctuations is required in order to obtain relaxation.
