  We explore dynamics of disordered and quasi-periodic interacting lattice
models using a self-consistent time-dependent Hartree-Fock (TDHF)
approximation, accessing both large systems (up to $L = 400$ sites) and very
long times (up to $t = 10^5$). We find that, in the $t \to \infty$ limit, the
many-body localization (MBL) is always destroyed within the TDHF approximation.
At the same time, this approximation provides important information on the
long-time character of dynamics in the ergodic side of the MBL transition.
Specifically, for one-dimensional (1D) disordered chains, we find slow
power-law transport up to the longest times, supporting the rare-region
(Griffiths) picture. The information on this subdiffusive dynamics is obtained
by the analysis of three different observables - temporal decay $\sim
t^{-\beta}$ of real-space and energy-space imbalances as well as domain wall
melting - which all yield consistent results. For two-dimensional (2D) systems,
the decay is faster than a power law, in consistency with theoretical
predictions that $\beta$ grows as $\log t$ for the decay governed by rare
regions. At longest times and moderately strong disorder, $\beta$ approaches
the limiting value $\beta=1$ corresponding to 2D diffusion. In quasi-periodic
(Aubry-Andr\'e) 1D systems, where rare regions are absent, we find considerably
faster decay that reaches the ballistic value $\beta=1$, which provides further
support to the Griffiths picture of the slow transport in random systems.
