Various elementary (de-) hydrogenation reactions at transition metal surfaces in heterogeneous catalysis have shown a linear Bronsted-Evans-Polanyi relation (BEP) between the activation energy and the reaction energy across metal surfaces. Using Density Functional Theory (DFT), we investigate if these BEP relations can be extended to elementary electrochemical reduction reactions involving transfer of a proton-electron pair. We examine the effect of the applied electrode potential on the BEP relations. We focus in particular on elementary electrochemical C-H, O-H and N-H bond formation reactions on 7 close-packed transition metal surfaces, including Ag (111), Au (111), Cu (111), Ni (111), Pt (111), Pd (111) and Rh (111). The potential-dependent activation energies used to construct the BEP relations are calculated using a Marcus theory based approach. The role of interfacial water in the kinetics and the reaction mechanism of C-H, O-H and N-H bond formation is explored. Specifically, two mechanisms are considered, a Tafel-like scheme involving direct surface hydrogenation and a Heyrovsky-like mechanism in which the proton is shuttled via an explicit water molecule. The potential-dependent elementary kinetics across three reduction reaction series, C* -> CH4, O* -> H2O and N* -> NH3, are also discussed. The Heyrovsky-like scheme is the preferred mechanism for C-H, O-H and N-H bond formation on all but two metals. BEP relations hold at the same applied electrode potential for C-H, O-H and N-H formation for both examined mechanisms. However, BEP parameters differ among C-H, O-H, and N-H reactions, indicating that elementary reaction energies alone are not sufficient for comparing the activity of catalysts when electrocatalytic reactions involve such different elementary steps.