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 intraspin coherences. Our predictions are readily testable in current experiments with ultracold alkaline-earth-metal(-like) atoms.