  In addition to the well-known $B \to D^{(*)} \ell \nu_\ell$ decays,
semileptonic $B_s \to D_s^{(*)} \ell \nu_\ell$ processes offer the possibility
to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\vert
V_{cb}\vert$. We implement the Dispersive Matrix (DM) approach to describe the
hadronic Form Factors (FFs) for the $B_s \to D_s^{(*)}$ transition in the whole
kinematical range, starting from recent Lattice QCD computations at large
values of the 4-momentum transfer. We extract $\vert V_{cb} \vert$ from the
experimental data, obtaining $\vert V_{cb} \vert \cdot 10^3 = (41.7 \pm 1.9)$
from $B_s \to D_s \ell \nu_\ell$ and $\vert V_{cb} \vert \cdot 10^3 =(40.7 \pm
2.4)$ from $B_s \to D_s^* \ell \nu_\ell$ decays. After averaging with the
values of $\vert V_{cb} \vert$ obtained from the $B \to D^{(*)}$ channels [1,2]
we get $\vert V_{cb} \vert \cdot 10^3 =(41.2 \pm 0.8)$, which is compatible
with the most recent inclusive estimate $\vert V_{cb} \vert_{\rm{incl}} \cdot
10^3 = 42.16 \pm 0.50$ [3] at the $1 \sigma$ level. In addition we test the
Lepton Flavour Universality (LFU) by computing the $\tau / \ell$ ratios of the
total decay rates (where $\ell$ is a light lepton), obtaining $R(D_s) =
0.298\,(5)$ and $R(D_s^*)= 0.250\,(6)$. We also address the issue of the
$SU(3)_F$ symmetry breaking by comparing the hadronic FFs entering the
semileptonic $B \to D^{(*)}$ and $B_s \to D_s^{(*)}$ channels.
