  Motivated by recent progress in cold-atom experiments, we analyze the SU($N$)
Fermi-Hubbard model on a $d$-dimensional hypercubic lattice with two-body loss.
By focusing on states near the ferromagnetic steady states, we obtain the
Liouvillian gap in closed form for any $d$ and $N$. We also investigate the
dynamics of a ferromagnetic initial state with a single spin flip both
analytically and numerically. In particular, we show that, by decreasing the
strength of the interaction and loss, the survival probability of the spin flip
exhibits a crossover from the power-law decay to the exponential decay. We
expect that our findings can be tested experimentally with ultracold
alkaline-earth-like atoms in an optical lattice.
