  We study limits from dilepton searches on leptoquark completions to the
Standard Model in the parameter space motivated by anomalies in the $b
\rightarrow s$ sector. After a full Run-2 analysis by LHCb, the disparity in
lepton flavour violation has disappeared. However, the mismatch in angular
distributions as well as in $B_s \rightarrow \mu^+ \mu^-$ partial width is
still unresolved and still implies a possible new physics contribution. We
probe three models of leptoquarks -- scalar models $S_3$ and $R_2$ as well as
vector leptoquark model $U_1$ using non-resonant dilepton searches to place
limit on both the mass and couplings to SM fermions. The exclusions of
leptoquarks coupling either non-uniformly to different lepton flavours or
uniformly is examined. Interestingly, if leptoquark couplings to electrons and
muons are indeed universal, then the $U_1$ model parameter space that
corresponds to the anomalous contribution should already accessible with Run-2
data in the non-resonant $e\mu$ channel. In the non-universal case, there is a
significant exclusion in couplings, but not enough to reach regions that
explain observed anomalies. We, therefore, examine the prospective sensitivity
at the HL-LHC as well as of a 3 TeV future muon collider. For the vector
leptoquark model, we find that a muon collider can probe all of the relevant
parameter space at 95% confidence with just 1 fb$^{-1}$ data whereas $R_2$ and
$S_3$ models can be excluded at 95% with 5 fb$^{-1}$ and 6.5 fb$^{-1}$
luminosity respectively.
