  We study the patterns of flavour violation in renormalisable extensions of
the Standard Model (SM) that contain vector-like quarks (VLQs) in a single
complex representation of either the SM gauge group $G_{SM}$ or $G_{SM}' =
G_{SM} \times U(1)_{L_\mu - L_\tau}$. We first decouple VLQs in the $M=(1-10)$
TeV range and then at the electroweak scale also $Z, Z'$ gauge bosons and
additional scalars to study the resulting phenomenology that depends on the
relative size of $Z$- and $Z'$-induced flavour-changing neutral currents, as
well as the size of $\Delta F=2$ contributions including the effects of
renormalisation group Yukawa evolution from $M$ to the electroweak scale that
turn out to be very important for models with right-handed currents through the
generation of left-right operators. In addition to rare decays like $P\to
\ell\bar\ell$, $P\to P' \ell\bar\ell$, $P\to P'\nu\bar\nu$ with $P=K, B_s, B_d$
and $\Delta F=2$ observables we analyze the ratio $\varepsilon'/\varepsilon$
which appears in the SM to be significantly below the data. We study patterns
and correlations between these observables which taken together should in the
future allow for differentiating between VLQ models. In particular the patterns
in models with left-handed and right-handed currents are markedly different
from each other. Among the highlights are large $Z$-mediated new physics
effects in Kaon observables in some of the models and significant effects in
$B_{s,d}$-observables. $\varepsilon'/\varepsilon$ can easily be made consistent
with the data, implying then uniquely the suppression of $K_L\to \pi^0
\nu\bar\nu$. Significant enhancements of $Br(K^+\to \pi^+ \nu\bar\nu)$ are
still possible. We point out that the combination of $\Delta F=2$ and $\Delta
F=1$ observables in a given meson system generally allows to determine the
masses of VLQs independently of the size of VLQ couplings.
