<p>We study, from a view of variational convergence, approximations of weak and strong set-valued quasi-variational inequalities and traffic network problems with arc capacity constraints, set-valued travel costs, and elastic demands depending on equilibrium flows. We define a bifunction associated to the considered problem and show that when such bifunctions of problems approximating this problem converge in appropriate types of variational convergence to the associated bifunction of the original problem, their approximate solutions metrically converge to solutions of the original problem. Here, we propose some new concepts of approximate solutions, saturatedness of arcs and paths, and equilibrium flows. The paper is the first attempt to consider global approximations of the aforementioned optimization models in terms of variational convergence. Hence, the novelty of results is high and they suggest further developments of the topic.</p>