  The photospheres of low-mass red giants show CNO isotopic abundances that are
not satisfactorily accounted for by canonical stellar models. The same is true
for the measurements of these isotopes and of the $^{26}$Al/$^{27}$Al ratio in
presolar grains of circumstellar origin. Non-convective mixing, occurring
during both Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stages is
the explanation commonly invoked to account for the above evidence. Recently,
the need for such mixing phenomena on the AGB was questioned, and chemical
anomalies usually attributed to them were suggested to be formed in earlier
phases. We have therefore re-calculated extra-mixing effects in low mass stars
for both the RGB and AGB stages, in order to verify the above claims. Our
results contradict them; we actually confirm that slow transport below the
convective envelope occurs also on the AGB. This is required primarily by the
oxygen isotopic mix and the $^{26}$Al content of presolar oxide grains. Other
pieces of evidence exist, in particular from the isotopic ratios of carbon
stars of type N, or C(N), in the Galaxy and in the LMC, as well as of SiC
grains of AGB origin. We further show that, when extra-mixing occurs in the RGB
phases of population I stars above about 1.2 $M_{\odot}$, this consumes $^3$He
in the envelope, probably preventing the occurrence of thermohaline diffusion
on the AGB. Therefore, we argue that other extra-mixing mechanisms should be
active in those final evolutionary phases.
