We study the variational convergence of robust integral functionals induced by empirical probability measures. We establish a generalized consistency framework where the ambiguity set is constructed using a probability metric. By replacing traditional uniform equicontinuity assumptions with inherent convexity and general probability metric domination, we prove the almost sure pointwise and uniform convergence of the robust objective and constraint functionals via convex analysis. Furthermore, using variational analysis, we demonstrate the Painlev\’e–Kuratowski set convergence of the empirical feasible regions and establish both optimal value convergence and the outer semicontinuity of the constrained minimizer sets.