  Evidence for electron-muon universality violation that has been revealed in
$b\to s \ell\ell$ transitions in the observables $R_{K,K^*}$ by the LHCb
Collaboration can be explained with spin-1 leptoquarks in $SU(2)_L$ singlet
$V_1$ or triplet $V_3$ representations in the ${\cal{O}}(1-10)$ TeV range. We
explore the sensitivity of the high luminosity LHC (HL-LHC) and future
proton-proton colliders to $V_1$ and $V_3$ in the parameter space connected to
$R_{K,K^*}$-data. We consider pair production and single production in
association with muons in different flavor benchmarks. Reinterpreting a recent
ATLAS search for scalar leptoquarks decaying to $b \mu$ and $j \mu$, we extract
improved limits for the leptoquark masses: For gauge boson-type leptoquarks
($\kappa=1$) we obtain $M_{V_1}> 1.9$ TeV, $M_{V_1}> 1.9$ TeV, and $M_{V_1}>
1.7$ TeV for leptoquarks decaying predominantly according to hierarchical,
flipped and democratic quark flavor structure, respectively. Future sensitivity
projections based on extrapolations of existing ATLAS and CMS searches are
worked out. We find that for $\kappa=1$ the mass reach for pair (single)
production of $V_1$ can be up to 3 TeV (2.1 TeV) at the HL-LHC and up to 15 TeV
(19.9 TeV) at the FCC-hh with $\sqrt{s}=100$ TeV and $20 \, \mbox{ab}^{-1}$.
The mass limits and reach for the triplet $V_3$ are similar or higher,
depending on flavor. While there is the exciting possibility that leptoquarks
addressing the $R_{K,K^*}$-anomalies are observed at the LHC, to fully cover
the parameter space $pp$-collisions beyond the LHC-energies are needed.
