Extra dimensions (ED) offer a valuable tool for constructing intricate models and exploring potential new physics phenomena. Our focus is to extend the Standard Model (SM) by introducing an \(U(1)_{L_{\mu } - L_{\tau }}\) gauge group in the framework of ED, which serves as a compelling initiative aimed at addressing the muon \((g - 2)\) anomaly. In this model, only the Kaluza-Klein (KK) modes of the extra-dimensional gauge boson traverse the bulk, while SM particles remain localised on a brane, assuming a compactification ‘radius’ of order \(R^{-1}\) form 1 to 1000 MeV. We present constraints on the \(m_\text {KK}\)  −  \(g'\) , where \(m_\text {KK}\) is the mass of the extra-dimensional gauge boson for the lowest KK mode, and \(g'\) is the interaction coefficient between the extra-dimensional gauge boson and SM fermions. These constraints are derived from \(\nu \)  − e elastic scattering measurements at DUNE Near Detector (ND), a future experiment, and compared with those from other \(\nu \)  − e elastic scattering experiments, CHARM-II, TEXONO and Borexino. Our results indicate that the DUNE ND will provide stringent bounds for extra-dimensional gauge boson masses in the 1 to 1000 MeV range.