Aims. We investigate the Na abundance distribution of asymptotic giant branch (AGB) stars in Galactic globular clusters (GCs) and its possible dependence on GC global properties, especially age and metallicity.Methods. We analyze high-resolution spectra of a large sample of AGB and red giant branch (RGB) stars in the Galactic GCs NGC 104, NGC 6121, and NGC 6809 obtained with FLAMES/GIRAFFE at ESO/VLT, and determine their Na abundances. This is the first time that the AGB stars in NGC 6809 are targeted. Moreover, to investigate the dependence of AGB Na abundance dispersion on GC parameters, we compare the AGB [Na/H] distributions of a total of nine GCs, with five determined by ourselves with homogeneous method and four from literature, covering a wide range of GC parameters.Results. 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 stars. By analyzing all nine GCs, we find that the Na abundances and multiple populations of AGB stars form 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 stars, and various cases exist between the two extremes. In addition, the fitted relations between fractions of the AGB second population and GC global parameters show that the AGB second-population fraction slightly anticorrelates with GC central concentration, while no robust dependency can be confirmed with other GC parameters.Conclusions. Current data roughly support the prediction of the fast-rotating massive star (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.