  We discuss Yukawa-enhanced contributions from $Z$-mediated new physics to
down-type quark $\Delta F=2$ processes in the framework of the standard model
gauge-invariant effective theory (SMEFT). Besides the renormalization group
(RG) mixing of the $Z$-mediating $\psi^2 H^2 D$ operators into $\Delta F = 2$
operators, we include at the electroweak scale one-loop (NLO) matching
corrections consistently, necessary for the removal of the matching scale
dependence. We point out that the right-handed $Z$-mediated interactions
generate through Yukawa RG mixing $\Delta F=2$ left-right operators, which are
further enhanced through QCD RG effects and chirally enhanced hadronic matrix
elements. We investigate the impact of these new effects on the known
correlations between $\Delta F=2$ and $\Delta F=1$ transitions in the SMEFT
framework and point out qualitative differences to previous parameterizations
of $Z$-mediated new physics that arise for the left-handed case. We illustrate
how specific models fit into our model-independent framework by using four
models with vector-like quarks. We carry out model-independent analyses of
scenarios with purely left-handed and purely right-handed new-physics $Z$
couplings for each of the three sectors $s\to d$, $b\to s$ and $b\to d$.
Specifically we discuss the correlations between $\varepsilon'/\varepsilon$,
$\varepsilon_K$, $K_L\to \mu^+\mu^-$ $K^+\to \pi^+\nu\bar\nu$ and
$K_L\to\pi^0\nu\bar\nu$ in the Kaon sector, and $\phi_s$, $B_s\to\mu^+\mu^-$
and $B\to K^{(*)} (\mu^+\mu^-, \nu\bar\nu)$ in the $b\to s$ sector and
$B_d\to\mu^+\mu^-$ in the $b\to d$ sector.
