We study the non-equilibrium dynamics of a disordered quantum system
consisting of harmonic oscillators in a $d$-dimensional lattice. If the system
is sufficiently localized, we show that, starting from a broad class of initial
product states that are associated with a tiling (decomposition) of the
$d$-dimensional lattice, the dynamical evolution of entanglement follows an
area law in all times. Moreover, the entanglement bound reveals a dependency on
how the subsystems are located within the lattice in dimensions $d\geq 2$. In
particular, the entanglement grows with the maximum degree of the dual graph
associated with the lattice tiling.
