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Context. Carbon-enhanced metal-poor (CEMP) stars are known to have properties that reflect the nucleosynthesis of the first low- and intermediate-mass stars, because most have been polluted by a now-extinct AGB star. Aims. By considering abundances in the various CEMP subclasses separately, we try to derive parameters (such as metallicity, mass, temperature, and neutron source) characterising AGB nucleosynthesis from the specific signatures imprinted on the abundances, and separate them from the impact of thermohaline mixing, first dredge-up, and dilution associated with the mass transfer from the companion. Methods. To place CEMP stars in a broader context, we collect abundances for about 180 stars of various metallicities (from solar to [Fe/H] $=-4$), luminosity classes (dwarfs and giants), and abundance patterns (e.g. C-rich and poor, Ba-rich and poor), from both our own sample and the literature. Results. We first show that there are CEMP stars that share the properties of CEMP-s stars and CEMP-no stars (which we refer to as CEMP-low-s stars). We also show that there is a strong correlation between Ba and C abundances in the s-only CEMP stars. This represents a strong detection of the operation of the $\rm^{13}$C neutron source in low-mass AGB stars. For the CEMP-rs stars (seemingly enriched with elements from both the s- and r-processes), the correlation of the N abundances with abundances of heavy elements from the 2nd and 3rd s-process peaks bears instead the signature of the $\rm^{22}$Ne neutron source. Since CEMP-rs stars also exhibit O and Mg enhancements, we conclude that extremely hot conditions prevailed during the thermal pulses of the contaminating AGB stars. We also note that abundances are not affected by the evolution of the CEMP-rs star itself (especially by the first dredge-up). This implies that mixing must have occurred while the star was on the main sequence, and that a large amount of matter must have been accreted so as to trigger thermohaline mixing. Finally, we argue that most CEMP-no stars (with neutron-capture element abundances comparable to non-CEMP stars) are likely the extremely metal-poor counterparts of CEMP neutron-capture-rich stars. We also show that the C enhancement in CEMP-no stars declines with metallicity at extremely low metallicity ([Fe/H] $< -3.2$). This trend is not predicted by any of the current AGB models.