Aims. We explore modifications to the current scenario for the slow neutron-capture process (the s-process) in asymptotic giant branch (AGB) stars to account for the Pb deficiency observed in post-AGB stars of low metallicity (Fe/H ≃−1.2) and low initial mass (≃ 1−1.5 M⊙) in the Large and Small Magellanic Clouds. Methods. We calculated the stellar evolution and nucleosynthesis for a 1.3 M⊙ star with Fe/H = −1.3 and tested different amounts and distributions of protons leading to the production of the main neutron source within the 13C-pocket and proton ingestion scenarios. Results. No s-process models can fully reproduce the abundance patterns observed in the post-AGB stars. When the Pb production is lowered, the abundances of the elements between Eu and Pb, such as Er, Yb, W, and Hf, are also lowered to below those observed. Conclusions. Neutron-capture processes with neutron densities intermediate between the s and the rapid neutron-capture processes may provide a solution to this problem and be a common occurrence in low-mass, low-metallicity AGB stars.