We examine the semileptonic $B \to D^{(*)} \ell \nu_\ell$ and $B \to \pi \ell \nu_\ell$ decays adopting the unitarity-based Dispersive Matrix (DM) method, which allows to determine the shape of the relevant hadronic form factors (FFs) in their whole kinematical range, using only lattice QCD results available at large values of the 4-momentum transfer without making any assumption on their momentum dependence. Moreover, the experimental data are not used to constrain the shape of the FFs, but only to obtain our final exclusive determination of $\vert V_{cb} \vert$ and $\vert V_{ub} \vert$, namely: $\vert V_{cb} \vert \cdot 10^3 = 41.1 \pm 1.0$ and $\vert V_{ub} \vert \cdot 10^3 = 3.88 \pm 0.32$, which are consistent with the latest inclusive determinations at the $1 \sigma$ level or better. Our calculation of the FFs allows to obtain pure theoretical estimates of the $\tau / \mu$ ratios of differential decay rates, $R(D) = 0.296 \pm 0.008$ and $R(D^*) = 0.275 \pm 0.008$, which turn out to be compatible with the experimental world averages within $\simeq 1.4$ standard deviations.