  We consider scenarios with a heavy $Z^\prime$ gauge boson with flavour
non-universal quark and lepton couplings with the goal to illustrate how the
cancellation of gauge anomalies generated by the additional U(1)$^\prime$ gauge
symmetry would imply correlations between FCNC processes in the quark sector,
in the lepton sector and most interestingly between quark flavour and lepton
flavour violating processes. We present scenarios with only left-handed
flavour-violating $Z^\prime$ couplings and those with also right-handed
flavour-violating couplings. These are characterized by a small number of free
parameters but in contrast to gauge anomaly cancellation in the Standard Model,
that takes place separately within each generation, in our scenarios anomaly
cancellation involves simultaneously quarks and leptons of all three
generations. Our models involve, beyond the ordinary quarks and leptons, three
heavy right-handed neutrinos. The models with only left-handed FCNCs of
$Z^\prime$ involve beyond $g_{Z^\prime}$ and $M_{Z^\prime}$ two real parameters
characterizing the charges of all fermions under the U(1)$^\prime$ gauge
symmetry and the CKM and PMNS ones in the quark and lepton sectors,
respectively. The models with the right-handed FCNCs of $Z^\prime$ involve few
additional parameters. Imposing constraints from well measured $\Delta F=2$
observables we identify interesting correlations that involve e.g.
$\varepsilon'/\varepsilon$, $B_{s,d}\to\mu^+\mu^-$, $B\to K(K^*) \ell^+\ell^-$,
$K^+\rightarrow\pi^+\nu\bar\nu$, $K_{L}\rightarrow\pi^0\nu\bar\nu$ and purely
lepton flavour violating decays like $\mu\to e\gamma$, $\mu\to 3 e$, $\tau\to
3\mu$ and $\mu-e$ conversion} among others. Also $(g-2)_{\mu,e}$ are
considered. The impact of the experimental $\mu\to e\gamma$, $\mu\to 3 e$ and
in particular $\mu-e$ conversion bounds on rare $K$ and $B$ decays is
emphasized.
