  We study the cosmological evolution of a FLRW universe dominated by the
energy density of moduli close to asymptotic regions of moduli space. Due to
the structure of the $\mathcal{N}=1$ SUGRA kinetic term, a saxion and an axion
residing in the same chiral multiplet are (universally) coupled even if the
latter is a flat direction of the potential, resulting in non-trivial dynamics.
We generalise known results in the literature to the case of multiple moduli,
showing the existence of various ``tracker" attractor solutions where the
relative energy densities of many components (axions included) stay in a fixed
ratio throughout the evolution. We conclude with some phenomenological
applications, relevant for both the early and late universe.
