  In this contribution we analyse the heavy-to-light $B$ decays through the
Dispersive Matrix method, which can be applied to any semileptonic decays of
hadrons once lattice QCD computations of the hadronic Form Factors and of the
relevant susceptibilities are available. We will explicitly discuss the
application of the Dispersive Matrix approach to both $B \to \pi \ell
\nu_{\ell}$ and $B_s \to K \ell \nu_{\ell}$ decays. As usual in our analysis
strategy, only LQCD computations of the FFs at high values of the momentum
transfer will be used to determine the shape of the FFs in the whole
kinematical range without making any assumption on their momentum dependence.
Then, the experimental data will be used only to obtain our final exclusive
determinations of $\vert V_{ub} \vert$. In this way, our calculation of the FFs
allows to obtain pure theoretical estimates of several quantities of
phenomenological interest, for instance the $\tau/\mu$ ratio of the
differential decay rates $R_{\pi(K)}^{\tau/\mu}$, which is an important tool
for testing Lepton Flavour Universality. We will also present a summary of all
the results obtained so far for semileptonic $B$ decays within the Dispersive
Matrix approach.
