We present computations of nucleosynthesis in low-mass (LM) red giant branch (RGB) and asymptotic giant branch (AGB) stars of Population I experiencing extended mixing. We adopt the updated version of the FRANEC evolutionary model, a new post-process code for non-convective mixing and the most recent revisions for solar abundances. In this framework, we discuss the effects of recent improvements in relevant reaction rates for proton captures on intermediate-mass (IM) nuclei (from carbon to aluminum). For each nucleus, we briefly discuss the new choices and their motivations. The calculations are then performed on the basis of a parameterized circulation, where the effects of the new nuclear inputs are best compared to previous works. We find that the new rates (and notably the one for the 14 N(p, γ ) 15 O reaction) imply considerable modifications in the composition of post-mainsequence stars. In particular, the slight temperature changes due to the reduced efficiency of proton captures on 14 N induce abundance variations at the first dredge-up (especially for 17 O, whose equilibrium ratio to 16 O is very sensitive to the temperature). In this new scenario, presolar oxide grains of AGB origin turn out to be produced almost exclusively by very low mass stars (M 1.5-1.7 M ), never becoming C-rich. The whole population of grains with 18 O/ 16 O below 0.0015 (the limit permitted by first dredge-up) is now explained. Also, there is now no forbidden area for very low values of 17 O/ 16 O (below 0.0005), contrary to previous findings. A rather shallow type of transport seems to be sufficient for the CNO changes in RGB stages. Both thermohaline diffusion and magnetic-buoyancy-induced mixing might provide a suitable physical mechanism for this. Thermohaline mixing is in any case certainly inadequate to account for the production of 26 Al on the AGB. Other transport mechanisms must therefore be at play. In general, observational constraints from RGB and AGB stars, as well as from presolar grains, are well reproduced by our approach. The nitrogen isotopic ratio in mainstream SiC grains remains an exception. For the low values measured in them (i.e., for 14 N/ 15 N 2000), we have no explanation. Actually, for the several grains with subsolar nitrogen isotopic ratios, no known stellar process acting in LM stars can provide a clue. This might be an evidence that some form of contamination from cosmic ray spallation occurs in the interstellar medium, adding fresh 15 N to the grains.