  Diboson production processes provide good targets for precision measurements
at present and future hadron colliders. We consider $Vh$ production, focusing
on the $h \to b\bar b$ decay channel, whose sizeable cross section makes it
accessible at the LHC. We perform an improved analysis by combining the 0-, 1-
and 2-lepton channels with a scale-invariant $b$-tagging algorithm that allows
us to exploit events with either a boosted Higgs via mass-drop tagging or
resolved $b$-jets. This strategy gives sensitivity to 4 dimension-6 SMEFT
operators that modify the $W$ and $Z$ couplings to quarks and is competitive
with the bounds obtained from global fits. The benefit of the $h\to b\bar b$
decay channel is the fact that it is the only $Vh$ channel accessible at the
LHC Run 3 and HL-LHC, while at FCC-hh it is competitive with the effectively
background-free $h\to \gamma\gamma$ channel assuming $\lesssim 5$% systematic
uncertainty. Combining the boosted and resolved categories yields a 17%
improvement on the most strongly bounded Wilson coefficient at the LHC Run 3
with respect to the boosted category alone (and a 7% improvement at FCC-hh). We
also show that, at FCC-hh, a binning in the rapidity of the $Vh$ system can
significantly reduce correlations between some EFT operators. The bounds we
obtain translate to a lower bound on the new physics scale of $5$, $8$, and
$20$ TeV at the LHC Run 3, HL-LHC, and FCC-hh respectively, assuming
new-physics couplings of order unity. Finally, we assess the impact of the $Vh$
production channel on anomalous triple gauge coupling measurements, comparing
with their determination at lepton colliders.
