We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.