  We study local quark-hadron duality and its violation for the
$D^0-\bar{D}^0$, $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings in the 't
Hooft model, offering a laboratory to test QCD in two-dimensional spacetime
together with the large-$N_c$ limit. With the 't Hooft equation being
numerically solved, the width difference is calculated as an exclusive sum over
two-body decays. The obtained rate is compared to inclusive one that arises
from four-quark operators to check the validity of the heavy quark expansion
(HQE). In view of the observation in four-dimensions that the HQE prediction
for the width difference in the $D^0-\bar{D}^0$ mixing is four orders of
magnitude smaller than the experimental data, in this work we investigate
duality violation in the presence of the GIM mechanism. We show that the order
of magnitude of the observable in the $D^0-\bar{D}^0$ mixing is enhanced in the
exclusive analysis relative to the inclusive counterpart, when the 4D-like
phase space function is used for the inclusive analysis. By contrast, it is
shown that for the $B^0_d-\bar{B}^0_d$ and $B^0_s-\bar{B}^0_s$ mixings, small
yet non-negligible corrections to the inclusive result emerge, which are still
consistent with what is currently indicated in four-dimensions.
