  We analyzed the high-resolution spectra of a large sample of AGB and RGB
stars in the Galactic globular clusters (GCs) NGC 104, NGC 6121, and NGC 6809
obtained with FLAMES/GIRAFFE at ESO/VLT, and determined their Na abundances.
This is the first time that the AGB stars in NGC 6809 are targeted. Moreover,
we reanalyzed with the same method the data archive for NGC 6752. To
investigate the dependence of AGB Na abundance dispersion on GC parameters, we
compared the AGB [Na/H] distributions we obtained (including NGC 2808 presented
in the first paper of this series) with literature data for four other GCs,
i.e., a total of nine GCs, covering a wide range of GC parameters. We find that
NGC 104 and NGC 6809 have comparable AGB and RGB Na abundance distributions
revealed by the K-S test, while NGC 6121 shows a lack of very Na-rich AGB star.
By analyzing all the nine GCs, the Na abundances and multiple populations of
AGB stars show a complex picture. In some GCs, AGB stars have similar Na
abundances and/or second-population fractions as their RGB counterparts, while
some GCs do not have Na-rich second-population AGB star, and various cases
exist between the two extremes. In addition, the fitted relations between
fractions of AGB second population and GC global parameters show that AGB
second-population fraction slightly anticorrelates with GC central
concentration, while no robust dependency can be confirmed with other GC
parameters. Current data roughly support the prediction of FRMS scenario.
However, considering the weak observational and theoretical trends where
scatter and exceptions exist, the fraction of second-population AGB stars can
be affected by more than one or two factors, and may even be a result of
stochasticity.
