  We calculate BSM hadronic matrix elements for $K^0-\bar K^0$ mixing in the
Dual QCD approach (DQCD). The ETM, SWME and RBC-UKQCD lattice collaborations
find the matrix elements of the BSM density-density operators $\mathcal{O}_i$
with $i=2-5$ to be rather different from their vacuum insertion values (VIA)
with $B_2\approx 0.5$, $B_3\approx B_5\approx 0.7$ and $B_4\approx 0.9$ at
$\mu=3~GeV$ to be compared with $B_i=1$ in the VIA. We demonstrate that this
pattern can be reconstructed within the DQCD through the non-perturbative meson
evolution from very low scales, where factorization of matrix elements is
valid, to scales of order $(1~GeV)$ with subsequent perturbative quark-gluon
evolution to $\mu=3~GeV$. This turns out to be possible in spite of a very
different pattern displayed at low scales with $B_2=1.2$, $B_3=3.0$, $B_4=1.0$
and $B_5\approx 0.2$ in the large $N$ limit, $N$ being the number of colours.
Our results imply that the inclusion of meson evolution in the phenomenology of
any non-leptonic transition like $K^0-\bar K^0$ mixing and $K\to\pi\pi$ decays
is mandatory. While meson evolution, as demonstrated in our paper, is hidden in
LQCD results, to our knowledge DQCD is the only analytic approach for
non-leptonic transitions and decays which takes this important QCD dynamics
into account.
