  We study the response of trapped two dimensional cold bosons to time
dependent lattices. We find that in lattice ramps from 11 (superfluid,
$\hbar/U_{\text{i}} = 3$ms, $\hbar/J_{\text{i}} = 45$ms) to 16 recoils (Mott,
$\hbar/U_{\text{f}} = 2$ms, $\hbar/J_{\text{f}} = 130$ms) the local number
fluctuations remains at their equilibrium values if ramps are slower than 3 ms.
Global transport, however, is much slower (1s), especially in the presence of
Mott shells. This separation of timescales has practical implications for cold
atom experiments and cooling protocols.
