  Non-equilibrium quantum phenomena are ubiquitous in nature. Yet, theoretical
predictions on the real-time dynamics of many-body quantum systems remain
formidably challenging, especially for high dimensions, strong interactions or
disordered samples. Here we consider a notable paradigm of strongly correlated
Fermi systems, the Mott phase of the Hubbard model, in a setup resembling
ultracold-gases experiments. We study the three-dimensional expansion of a
cloud into an optical lattice after removing the confining potential. We use
time-dependent density-functional theory combined with dynamical mean-field
theory, considering interactions below and above the Mott threshold, as well as
disorder effects. At strong coupling, we observe multiple timescales in the
melting of the Mott wedding-cake structure, as the Mott plateau persist orders
of magnitude longer than the band insulating core. We also show that disorder
destabilises the Mott plateau and that, compared to a clean setup, localisation
can decrease, creating an interesting dynamic crossover during the expansion.
