  Transport and the approach to equilibrium in interacting classical and
quantum systems is a challenging problem of both theoretical and experimental
interest. One useful organizing principle characterizing equilibration is the
dissipative universality class, the most prevalent one being diffusion. In this
paper, we use the effective field theory (EFT) of diffusion to systematically
obtain universal power-law corrections to diffusion. We then employ large-scale
simulations of classical and quantum systems to explore their validity. In
particular, we find universal scaling functions for the corrections to the
dynamical structure factor $\langle n(x,t)n\rangle$, in the presence of a
single $U(1)$ or $SU(2)$ charge in systems with and without particle-hole
symmetry, and present the framework to generalize the calculation to multiple
charges. Classical simulations show remarkable agreement with EFT predictions
for subleading corrections, pushing precision tests of effective theories for
thermalizing systems to an unprecedented level. Moving to quantum systems, we
perform large-scale tensor-network simulations in unitary and noisy 1d Floquet
systems with conserved magnetization. We find a qualitative agreement with EFT
which becomes quantitative in the case of noisy systems. Additionally, we show
how the knowledge of EFT corrections allows for fitting methods, which can
improve the estimation of transport parameters at the intermediate times
accessible by simulations and experiments. Finally, we explore non-linear
response in quantum systems and find that EFT provides an accurate prediction
for its behavior. Our results provide a basis for a better understanding of the
non-linear phenomena present in thermalizing systems.
