Long-lived or stable heavy multiply-charged particles X, predicted in various Beyond-the-Standard-Model (BSM) scenarios, can significantly affect Big Bang Nucleosynthesis (BBN). The neutralization process of such states via the capture of light primordial nuclei N (e.g., p, d, t, 3He, 4He) leads to the formation of bound states XN (electrically neutral dark atoms, negatively charged dark ions or positively charged anomalous isotopes) and a shift in the ratios of the primordial abundances of ordinary light elements. The dependence of reaction rates on the charge of the heavy particle and the deviations from the standard BBN predictions are studied using a combination of analytic and numerical estimates. The cross-section of the first stage of the dark recombination may be calculated numerically in the dipole approximation. The finite size of the nucleus in the shell of the dark atom is taken into account. The rates of further reactions are strongly affected by strong nuclear forces and can be estimated by scaling the experimental data for the corresponding ordinary nuclear fusion processes. The reduced Coulomb barrier and modified reduced masses are considered. Changes in the reaction network are predicted for high charges. To avoid contradictions with the observed ratios of the primordial abundances, the fine-tuning of the model parameters (charge, mass, and baryon-to-photon ratio) may be required.