  Assuming confirmation of the anomalies in semi-leptonic $B$-decays, their
explanation in terms of the exchange of a massive vector leptoquark field,
$U_\mu^a$, of charge 2/3, appears to require the inclusion of $U_\mu^a$ in the
vector multiplet of the adjoint of Pati-Salam $SU(4)$, $\mathcal{G}_\mu^A$, as
well as the introduction of vector-like fermions, $F_j$, in the fundamental of
$SU(4)$. We consider simplified models characterised by the nature of the
$SU(4)$ symmetry (global or local), by the number of vector-like fermions, and
by the couplings of the Higgs boson to SM fermions (direct or induced by mixing
with the vector-like fermions). In all cases, we implement a minimal breaking
of a $U(2)^n_f$ flavour symmetry, with a single motivated exception. We then
perform a global fit including the main observables sensitive to exchanges of
the $\mathcal{G}_\mu^A$ at tree level and in loops dominated by logs
insensitive to the UV completion.
