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We theoretically study the dynamics of n-level spin-orbit coupled alkaline-earth fermionic atoms with SU(n) symmetric interactions. We consider three-dimensional lattices with tunneling along one dimension, and the internal levels treated as a synthetic dimension, realizing an n-leg flux ladder. Laser driving is used to couple the internal levels and to induce an effective magnetic flux through the ladder. We focus on the dense and strongly interacting regime, where in the absence of flux the system behaves as a Mott insulator with suppressed motional dynamics. At integer and fractional ratios of the laser Rabi frequency to the onsite interactions, the system exhibits resonant features in the dynamics. These resonances occur when interactions help overcome kinetic constraints upon the tunneling of atoms, thus enabling motion. Different resonances allow for the development of complex chiral current patterns. The resonances resemble those appearing in the longitudinal Hall resistance when the magnetic field is varied. We characterize the dynamics by studying the system’s long-time relaxation behavior as a function of flux, number of internal levels n, and interaction strength. We observe a series of nontrivial prethermal plateaus caused by the emergence of resonant processes at successive orders in perturbation theory. We discuss protocols to observe the predicted phenomena under current experimental conditions.