  We investigate the behaviour of asymptotic giant branch (AGB) stars between
metallicities Z = 10-4 and Z = 10-8 . We determine which stars undergo an
episode of flash-driven mixing, where protons are ingested into the intershell
convection zone, as they enter the thermally pulsing AGB phase and which
undergo third dredge-up. We find that flash-driven mixing does not occur above
a metallicity of Z = 10-5 for any mass of star and that stars above 2 M do not
experience this phenomenon at any metallicity. We find carbon ingestion (CI),
the mixing of carbon into the tail of hydrogen burning region, occurs in the
mass range 2 M to around 4 M . We suggest that CI may be a weak version of the
flash-driven mechanism. We also investigate the effects of convective
overshooting on the behaviour of these objects. Our models struggle to explain
the frequency of CEMP stars that have both significant carbon and nitrogen
enhancement. Carbon can be enhanced through flash-driven mixing, CI or just
third dredge up. Nitrogen can be enhanced through hot bottom burning and the
occurrence of hot dredge-up also converts carbon into nitrogen. The C/N ratio
may be a good indicator of the mass of the primary AGB stars.
