  Extending the Standard Model with higher-dimensional operators in an
effective-field-theory (EFT) approach provides a systematic framework to study
new-physics (NP) effects from a bottom-up perspective, as long as the NP scale
is sufficiently large compared to the energies probed in the experimental
observables. However, when taking into account the different quark and lepton
flavours, the number of free parameters increases dramatically, which makes
generic studies of the NP flavour structure infeasible. In this paper, we
address this issue in view of the recently observed "flavour anomalies" in
$B$-meson decays, which we take as a motivation to develop a general framework
that allows us to systematically reduce the number of flavour parameters in the
EFT. This framework can be easily used in global fits to flavour observables at
Belle II and LHCb as well as in analyses of flavour-dependent collider
signatures at the LHC. Our formalism represents an extension of the well-known
minimal-flavour-violation approach, and uses Froggatt-Nielsen charges to define
the flavour power-counting. As a relevant illustration of the formalism, we
apply it to the flavour structures which could be induced by a $U_1$ vector
leptoquark, which represents one of the possible explanations for the recent
hints of flavour non-universality in semileptonic $B$-decays. We study the
phenomenological viability of this specific framework performing a fit to
low-energy flavour observables.
