A matter of utmost focus in experimental and theoretical research in high energy physics today is testing the Standard Model through searches of deviations from its predictions. The determination of Cabibbo-Kobayashi-Maskawa quark mixing matrix is one of the key testing grounds in particle physics phenomenology. This is especially the case for the |V_ub| matrix element, where a tension has been noticed between the values obtained in inclusive and exclusive determinations. In recent years, a deviation of measurements of several observables in semileptonic decays of heavy mesons from their expected Standard Model values has sparked a renewed interest in theoretical research of what is known as lepton flavour universality, or the universality of weak coupling constant among lepton flavours. Since the study of quark flavour transitions necessarily deals with hadronic physics involving strong, non-perturbative interactions which introduce large uncertainties, a large effort has been made to put these uncertainties under control. At the moment, there are several theoretical tools at disposal used to extract the values of desired hadronic parameters: lattice quantum chromodynamics, effective theories and various types of sum rules based on quark-hadron duality. In this thesis, sum rules will be used in a computation of Bc and Bu meson transition form factors. Extracting non-perturbative quantities will enable scrutinizing SM predictions of important semileptonic B_c and B_u decays. In particular, an extraction of the |V_ub| CKM matrix element from B̅ → πlv̅_l and B_c → D^(*)lv̅_l decays is performed, alongside a test of lepton flavour universality in the SM. In the B_c → (J/ψ,η_c)lv̅_l decays new physics effects are additionally explored.