  We present the first results of our ongoing chemical study of carbon stars in
the Local Group of galaxies. We used spectra obtained with UVES at the 8.2 m
Kueyen-VLT telescope and a new grid of spherical model atmospheres for cool
carbon-rich stars which include polyatomic opacities, to perform a full
chemical analysis of one carbon star, BMB-B~30, in the Small Magellanic Cloud
(SMC) and two, IGI95-C1 and IGI95-C3, in the Sagittarius Dwarf Spheroidal (Sgr
dSph) galaxy. Our main goal is to test the dependence on the stellar
metallicity of the s-process nucleosynthesis and mixing mechanism occurring in
AGB stars. For these three stars, we find important s-element enhancements with
respect to the mean metallicity ([M/H]), namely [s/M]$\approx$+1.0, similar to
the figure found in galactic AGB stars of similar metallicity. The abundance
ratios derived between elements belonging to the first and second s-process
abundance peaks, corresponding to nuclei with a magic number of neutrons N=50
(88Sr, 89Y, 90Zr) and N=82 (138Ba, 139La, 140Ce, 141Pr), agree remarkably well
with the theoretical predictions of low mass (M $<3$~M$\_\odot$) metal-poor AGB
nucleosynthesis models where the main source of neutrons is the
$^{13}$C$(\alpha,n)^{16}$O reaction activated duringthe long interpulse phase,
in a small pocket located within the He-rich intershell. The derived C/O and
$^{12}$C/$^{13}$C ratios are, however, more difficult to reconcile with
theoretical expectations. Possible explanations, like the extrinsic origin of
the composition of these carbon stars or the operation of a non-standard mixing
process during the AGB phase (such as the {\it cool bottom process}), are
discussed on the basis of the collected observational constraints.
