Context. This paper is part of a larger project in which we systematically study the chemical abundances of Galactic and extragalactic post-asymptotic giant branch (post-AGB) stars. The goal at large is to provide improved observational constraints to the models of the complex interplay between the AGB s-process nucleosynthesis and the associated mixing processes. Aims. Lead (Pb) is the final product of the s-process nucleosynthesis and is predicted to have large overabundances with respect to other s-process elements in AGB stars of low metallicities. However, Pb abundance studies of s-process enriched post-AGB stars in the Magellanic Clouds show a discrepancy between observed and predicted Pb abundances. The determined upper limits based on spectral studies are much lower than what is predicted. In this paper, we focus specifically on the Pb abundance of 14 Galactic s-process enhanced post-AGB stars to check whether the same discrepancy is present in the Galaxy as well. Among these 14 objects, two were not yet subject to a detailed abundance study in the literature. We apply the same method to obtain accurate abundances for the 12 others. Our homogeneous abundance results provide the input of detailed spectral synthesis computations in the spectral regions where Pb lines are located. Methods. We used high-resolution UVES and HERMES spectra for detailed spectral abundance studies of our sample of Galactic post-AGB stars. None of the sample stars display clear Pb lines, and we only deduced upper limits of the Pb abundance by using spectrum synthesis in the spectral ranges of the strongest Pb lines. Results. We do not find any clear evidence of Pb overabundances in our sample. The derived upper limits are strongly correlated with the effective temperature of the stars with increasing upper limits for increasing effective temperatures. We obtain stronger Pb constraints on the cooler objects. Moreover, we confirm the s-process enrichment and carbon enhancement of two unstudied 21 μm sources IRAS 13245-6428 and IRAS 14429-4539. The mildly s-process enhanced post-AGB star IRAS 17279-1119 is part of a binary system and may be the long sought precursor of extrinsic Ba stars. Conclusions. Stars with Teff> 7500 K do not provide strong constraints on the Pb abundance as the strongest line in the optical spectrum is only detectable at unrealistically high Pb atmospheric abundances. Combining the Pb abundance results from this study with abundances from our previous studies, we conclude that the discrepancy between theory and observation increases towards lower metallicities. The model predictions are consistent with the deduced upper limits on the Pb abundances for all stars with Fe/H >−0.7 dex. For stars with Fe/H <−0.7 dex, however, the model predictions overestimate the Pb abundances with respect to the other s-process elements. All objects, except IRAS 17279-1119, confirm the relation between neutron exposure hs/ls and third dredge-up efficiency s/Fe, whereas no relation between metallicity and neutron exposure is detected within the metallicity range of our total sample (−1.4 < Fe/H < −0.2). The mild enrichment of IRAS 17279-1119 can probably be attributed to a cut-off of the AGB evolution due to binary interactions. To our knowledge, IRAS 17279-1119 is the first s-process enhanced Galactic post-AGB star known in a binary system and is a possible precursor of the extrinsic Ba dwarf stars. We corroborate the finding that the variety in abundance profiles shows that a large spread of neutron irradiation is needed for a given metallicity. Lead-rich stars are yet to be found among post-AGB stars.