Many of the carbon-enhanced metal-poor (CEMP) stars that we observe in the Galactic halo are found in binary systems and show enhanced abundances of elements produced by the slow neutron-capture process (s-elements). The origin of the peculiar chemical abundances of these CEMP-s stars is believed to be accretion in the past of enriched material from a primary star in the asymptotic giant branch (AGB) phase of its evolution. We investigate the mechanism of mass transfer and the process of nucleosynthesis in low-metallicity AGB stars by modelling 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 of binary stars generated by our binary evolution and nucleosynthesis model. We classify our sample CEMP-s stars in three groups based on the observed abundance of europium. In CEMP-s/r stars the europium-to-iron ratio is more than ten times higher than in the Sun, whereas it is lower than this threshold in CEMP-s/nr stars. No measurement of europium is currently available for CEMP-s/ur stars. On average our models reproduce the abundances observed in CEMP-s/nr stars well, whereas in CEMP-s/r stars and CEMP-s/ur stars the abundances of the light-s elements (strontium, yttrium, zirconium) are systematically overpredicted by our models, and in CEMP-s/r stars the abundances of the heavy-s elements (barium, lanthanum) are underestimated. In all stars our modelled abundances of sodium overestimate the observations. This discrepancy is reduced only in models that underestimate the abundances of most of the s-elements. Furthermore, the abundance of lead is underpredicted in most of our model stars, independent of the metallicity. These results point to the limitations of our AGB nucleosynthesis model, particularly in the predictions of the element-to-element ratios. In our models CEMP-s stars are typically formed in wide systems with periods above 10000-days, while most of the observed CEMP-s stars are found in relatively close orbits with periods below 5000-days. This evidence suggests that either the sample of CEMP-s binary stars with known orbital parameters is biased towards short periods or that our wind mass-transfer model requires more efficient accretion in close orbits.