Semileptonic heavy-to-heavy decays are very intriguing transitions because of the long-standing tension between the inclusive and the exclusive determinations of the Cabibbo-Kobayashi-Maskawa matrix element . Another discrepancy exists between the Standard Model (SM) expectations and the measurements of the ratios , which are a test of Lepton Flavour Universality. In both cases, a central role is played by the hadronic Form Factors (FFs) describing these decays. My goal is to review the quark transitions through the novel Dispersive Matrix approach, which is based on the non-perturbative determination of the dispersive bounds. I will describe how to obtain new values for and from semileptonic and decays. Particular attention will be devoted to the semileptonic channel, which is the main source of the aforementioned tensions. A critical comparison with the results from the Unitarity Triangle Analysis and the ones from HFLAV SM predictions will be then discussed. Finally, I will discuss in detail the state-of-the-art of the computations of the FFs on the lattice and their comparison with experimental measurements.