We study the slow relaxation of isolated quasi-integrable systems, focusing on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is well known that the initial energy sharing between different linear modes can be inferred by the integrable Toda chain. Using numerical simulations, we show explicitly how the relaxation of the FPU chain toward equilibration is determined by a slow drift within the space of Toda's integrals of motion. We analyze the whole spectrum of Toda modes and show how they dictate, via a generalized Gibbs ensemble, the quasistatic states along the FPU evolution. This picture is employed to devise a fast numerical integration, which can be generalized to other quasi-integrable models. In addition, we discuss how a fluctuation theorem, recently derived in Goldfriend and Kurchan [Europhys. Lett. 124, 10002 (2018)], describes the large deviations as the system flows in the entropy landscape.