  In recent times, one of the strongest hints of Physics Beyond the Standard
Model (BSM) has been the anomaly in the ratios $R_D$ and $R_{D^\ast}$ measured
in the charged current decays of $B$-mesons. In this work, we perform a
comprehensive analysis of these decay modes, first in a model independent way
and subsequently, in the context of composite Higgs models. We discuss in depth
as to how linearly realised $\rm SU(2)_L \times U(1)_Y$ symmetry imposes severe
constraint on the various scenarios because of correlations with other $\Delta
F =1$ processes and $Z\,\tau\,\tau$ and $Z\,\nu\,\nu$ couplings. In the
composite Higgs paradigm with partial compositeness, we show that, irrespective
of the flavour structure of the composite sector, constraints from $\Delta F
=2$ processes bring the compositeness scale down to $\sim 650$ GeV which is in
tension with electroweak precision observables. In the presence of composite
leptoquarks, the situation improves only marginally (a factor of $\sqrt{2}$ in
the compositeness scale), thus making the new states soon discoverable by
direct searches at the LHC. We also comment on the possible explanation of the
$R_{K,K^*}$ anomalies within the composite Higgs framework.
