  We study multilevel fermions in an optical lattice described by the Hubbard
model with on site SU($n$)-symmetric interactions. We show that in an
appropriate parameter regime this system can be mapped onto a spin model with
all-to-all SU($n$)-symmetric couplings. Raman pulses that address internal spin
states modify the atomic dispersion relation and induce spin-orbit coupling,
which can act as a synthetic inhomogeneous magnetic field that competes with
the SU($n$) exchange interactions. We investigate the mean-field dynamical
phase diagram of the resulting model as a function of $n$ and different initial
configurations that are accessible with Raman pulses. Consistent with previous
studies for $n=2$, we find that for some initial states the spin model exhibits
two distinct dynamical phases that obey simple scaling relations with $n$.
Moreover, for $n>2$ we find that dynamical behavior can be highly sensitive to
initial intra-spin coherences. Our predictions are readily testable in current
experiments with ultracold alkaline-earth(-like) atoms.
