We present models for the slow neutron-capture process (s-process) in asymptotic giant branch stars of metallicity [Fe/H] = -2.3 and masses 0.9-6 M-circle dot. We encountered different regimes of neutron-capture nucleosynthesis listed here increasing in importance as the stellar mass decreases: the Ne-22(alpha, n)Mg-25 reaction activated during the thermal pulses (TPs), the C-13(alpha, n)O-16 reaction activated in radiative conditions during the interpulse periods, and the C-13(alpha, n)O-16 reaction activated during the TPs, also as a result of mild proton-ingestion episodes. The models where the C-13 burns radiatively (masses similar or equal to 2 M-circle dot) produce an overall good match to carbon-enhanced metal-poor (CEMP) stars showing s-process enhancements (CEMP-s), except they produce too much Na and F. On the other hand, none of our models can provide a match to the composition of CEMP stars also showing rapid-process enhancements (CEMP-s/r). The models fail to reproduce the observed Eu abundances, and they also fail to reproduce the correlation between the Eu and Ba abundances. They also cannot match the ratio of heavy-to-light s-process elements observed in many CEMP-s/r stars, which can be more than 10 times higher than in the solar system. To explain the composition of CEMP-s/r stars we need to invoke the existence of a different s/r neutron-capture process either with features in between the s- and the r-processes, or generated by superpositions of different neutron-capture processes in the same astrophysical site or in sites linked to each other-for example, in multiple stellar systems.