  We develop a numerical method for the time evolution of Gaussian wave packets
on flat-band lattices in the presence of correlated disorder. To achieve this,
we introduce a method to generate random on-site energies with prescribed
correlations. We verify this method with a one-dimensional (1D) cross-stitch
model, and find good agreement with analytical results obtained from the
disorder-dressed evolution equations. This allows us to reproduce previous
findings, that disorder can mobilize 1D flat-band states which would otherwise
remain localized. As explained by the corresponding disorder-dressed evolution
equations, such mobilization requires an asymmetric disorder-induced coupling
to dispersive bands, a condition that is generically not fulfilled when the
flat-band is resonant with the dispersive bands at a Dirac point-like crossing.
We exemplify this with the 1D Lieb lattice. While analytical expressions are
not available for the two-dimensional (2D) system due to its complexity, we
extend the numerical method to the 2D $\alpha-T_3$ model, and find that the
initial flat-band wave packet preserves its localization when $\alpha = 0$,
regardless of disorder and intersections. However, when $\alpha\neq 0$, the
wave packet shifts in real space. We interpret this as a Berry phase
controlled, disorder-induced wave-packet mobilization. In addition, we present
density functional theory calculations of candidate materials, specifically
$\rm Hg_{1-x}Cd_xTe$. The flat-band emerges near the $\Gamma$ point
($\bf{k}=$0) in the Brillouin zone.
