  We study the real-time dynamics of a two-dimensional Anderson--Hubbard model
using nonequilibrium self-consistent perturbation theory within the second-Born
approximation. When compared with exact diagonalization performed on small
clusters, we demonstrate that for strong disorder this technique approaches the
exact result on all available timescales, while for intermediate disorder, in
the vicinity of the many-body localization transition, it produces
quantitatively accurate results up to nontrivial times. Our method allows for
the treatment of system sizes inaccessible by any numerically exact method and
for the complete elimination of finite size effects for the times considered.
We show that for a sufficiently strong disorder the system becomes nonergodic,
while for intermediate disorder strengths and for all accessible time scales
transport in the system is strictly subdiffusive. We argue that these results
are incompatible with a simple percolation picture, but are consistent with the
heuristic random resistor network model where subdiffusion may be observed for
long times until a crossover to diffusion occurs. The prediction of slow
finite-time dynamics in a two-dimensional interacting and disordered system can
be directly verified in future cold atoms experiments
