Nonequilibrium effects in interacting systems are among the most challenging problems in condensed-matter physics. Here, we present a quantum Hall-based setup where a simple thermoelectrical DC measurement extract information on the interaction-mediated thermal equilibration processes. We show how thermal relaxation between edge states, initially at different temperatures, evolves in terms of the interaction strength, the length of the interaction region and the temperature differences, finding a universal behavior. We compare the predictions between Landau-Fermi liquid and Tomonaga-Luttinger liquid. Finally, we apply a new circuit theory demonstrating how the cross-correlations crucially affect the energy relaxation process and the electron-distribution evolution. [Full article]