Starting from the formally exact Liouville equation, we derive a generalized Langevin equation for the simultaneous correlated motion of several molecules in dense melts. Our calculations are based upon phase space kinetic theory and Mori-Zwanzig projection operator techniques. In the overdamped regime, a set of nonlinear coupled equations is found containing cross-interaction contributions to the frequency matrix and to the memory functions. Specializing the problem to the correlated dynamics of polymer melts enables an analytical solution for the nonlinear cross contribution of the frequency matrix. The memory functions are calculated using a generalization of the Schweizer's single molecule mode-coupling approach. A self-consistent procedure allows a numerical solution of the diffusive dynamics of the chains on the scale of the intermolecular interactions. At long time, depending on the strength of the intermolecular mean-force potential, two different scenarios take place. For weak intermolecular interactions the short-time correlated diffusive dynamics crosses over to the uncorrelated single chain dynamics. For stronger intermolecular interactions, when the time scale of relaxation of the many chain domains exceeds the time scales investigated, collective diffusion dominates the dynamical behavior.