Stellar evolution models predict that internal mixing should cause some sodium overabundance at the surface of red giants more massive than ~ 1.5--2.0 Msun. The surface aluminium abundance should not be affected. Nevertheless, observational results disagree about the presence and/or the degree of the Na and Al overabundances. In addition, Galactic chemical evolution models adopting different stellar yields lead to quite different predictions for the behavior of [Na/Fe] and [Al/Fe] versus [Fe/H]. Overall, the observed trends of these abundances with metallicity are not well reproduced. We readdress both issues, using new Na and Al abundances determined within the Gaia-ESO Survey, using two samples: i) more than 600 dwarfs of the solar neighborhood and of open clusters and ii) low- and intermediate-mass clump giants in six open clusters. Abundances of Na in giants with mass below ~2.0 Msun, and of Al in giants below ~3.0 Msun, seem to be unaffected by internal mixing processes. For more massive giants, the Na overabundance increases with stellar mass. This trend agrees well with predictions of stellar evolutionary models. Chemical evolution models that are able to fit well the observed [Na/Fe] vs. [Fe/H] trend in solar neighborhood dwarfs can not simultaneously explain the run of [Al/Fe] with [Fe/H], and viceversa. The comparison with stellar ages is hampered by severe uncertainties. Indeed, reliable age estimates are available for only a half of the stars of the sample. We conclude that Al is underproduced by the models, except for stellar ages younger than about 7 Gyr. In addition, some significant source of late Na production seems to be missing in the models. Either current Na and Al yields are affected by large uncertainties, and/or some important Galactic source(s) of these elements has not been taken into account up to now. [abridged]