We consider N uniformly accelerating Unruh-DeWitt detectors whose internal degrees of freedom are coupled to a massless scalar field in (1 + 1)D Minkowski space. We use the influence functional formalism to derive the Langevin equations governing the nonequilibrium dynamics of the internal degrees of freedom and show explicitly that the system relaxes in time and equilibrates. We also show that once the equilibrium condition is established a set of fluctuation-dissipation relations (FDRs) and correlation-propagation relations emerges for the detectors, extending earlier results of Raval, Hu, and Anglin [Stochastic theory of accelerated detectors in quantum fields, Phys. Rev. D 53, 7003 (1996)] which discovered these relations for the quantum field. Although similar in form to the FDRs commonly known from linear response theory, which assumes an equilibrium condition a priori, their physical connotations are dissimilar from that of a nonequilibrium origin. We show explicitly that both sets of relations are needed to guarantee the balance of energy flow in and out of the system in dynamical equilibrium with the field. These results are helpful to investigations of quantum information and communications of detectors in space experiments and inquiries of theoretical issues in black holes and cosmology.