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Many observed CEMP stars are found in binary systems and show enhancedabundances of $s$-elements. The origin of the chemical abundances of theseCEMP-$s$ stars is believed to be accretion in the past of enriched materialfrom a primary star in the AGB phase. We investigate the mechanism of masstransfer and the process of nucleosynthesis in low-metallicity AGB stars bymodelling the binary systems in which the observed CEMP-$s$ stars were formed.For this purpose we compare a sample of $67$ CEMP-$s$ stars with a grid ofbinary stars generated by our binary evolution and nucleosynthesis model. Weclassify our sample CEMP-$s$ stars in three groups based on the observedabundance of europium. In CEMP$-s/r$ stars the europium-to-iron ratio is morethan ten times higher than in the Sun, whereas it is lower than this thresholdin CEMP$-s/nr$ stars. No measurement of europium is currently available forCEMP-$s/ur$ stars. On average our models reproduce well the abundances observedin CEMP-$s/nr$ stars, whereas in CEMP-$s/r$ stars and CEMP-$s/ur$ stars theabundances of the light-$s$ elements are systematically overpredicted by ourmodels and in CEMP-$s/r$ stars the abundances of the heavy-$s$ elements areunderestimated. In all stars our modelled abundances of sodium overestimate theobservations. This discrepancy is reduced only in models that underestimate theabundances of most of the $s$-elements. Furthermore, the abundance of lead isunderpredicted in most of our model stars. These results point to thelimitations of our AGB nucleosynthesis model, particularly in the predictionsof the element-to-element ratios. Finally, in our models CEMP-$s$ stars aretypically formed in wide systems with periods above 10000 days, while most ofthe observed CEMP-$s$ stars are found in relatively close orbits with periodsbelow 5000 days.

Carbon-enhanced metal-poor stars: a window on AGB nucleosynthesis and binary evolution. II. Statistical analysis of a sample of 67 CEMP-$s$ stars