We present models for the slow neutron-capture process ( s process ) in asymptotic giant branch ( AGB ) stars of metallicity [ Fe/H ] = - 2.3 and masses 0.9 { M } _ { \sun } to 6 { M } _ { \sun } . We encountered different regimes of neutron-capture nucleosynthesis increasing in importance as the stellar mass decreases : the ^ { 22 } Ne ( \alpha , n ) ^ { 25 } Mg reaction activated during the thermal pulses , the ^ { 13 } C ( \alpha , n ) ^ { 16 } O reaction activated in radiative conditions during the interpulse periods , and the ^ { 13 } C ( \alpha , n ) ^ { 16 } O reaction activated during the thermal pulses , also as a result of mild proton ingestion episodes . The models where the ^ { 13 } C burns radiatively ( masses \simeq 2 { M } _ { \sun } ) 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 can not match the ratio of heavy to light s-process elements observed in many CEMP- s / r stars , which can be more than ten 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 .