We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi-variational inequalities. These equations are related to robust switching control problems introduced in [E. Bayraktar, A. Cosso, and H. Pham, SIAM J. Control Optim., 54 (2016), pp. 2594-2628]. We prove that, as the time horizon goes to infinity (resp., discount factor goes to zero), the long run average solution to the parabolic system (resp., the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any nondegeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.