Résumé : We present a combined experimental and theoretical study of the mutual neutralization (MN) process in collisions of lithium ions (Li+) with deuterium anions (D-) at collision energies below 1 eV. We employ a merged-beam apparatus to determine total and state-to-state MN cross sections. We perform nuclear dynamics calculations using the multichannel Landau-Zener model based on accurate ab initio molecular data. We obtain an excellent agreement between the experimental and theoretical results over the energy range covered in this work. We show that the basis sets used in the ab initio calculations have a limited influence on the total cross section, but strongly impacts the results obtained for the partial cross sections or the reaction branching ratios. This demonstrates the important role of high-precision measurements to validate the theoretical approaches used to study gas-phase reactive processes. Finally, we compute MN rate coefficients for Li+ + H- and Li+ + D-, and discuss their significance for astrochemistry models.