We present computations of nucleosynthesis in low-mass red-giant-branch and asymptotic-giant-branch 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 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 14N(p,g)15O reaction) imply considerable modifications in the composition of post-main sequence stars. In particular, the slight temperature changes due to the reduced efficiency of proton captures on 14N induce abundance variations at the first dredge up (especially for 17O, whose equilibrium ratio to 16O 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