We used FLAMES+GIRAFFE (Medusa mode) at the VLT to obtain moderately high resolution spectra for 30 red horizontal branch (RHB) stars, 4 RR Lyrae variables, and 17 blue horizontal branch (BHB) stars in the low-concentration, moderately metal-rich globular cluster NGC 6723 ([Fe/H] = -1.22 ± 0.08 from our present sample). The spectra were optimized to derive O and Na abundances. In addition, we obtained abundances for other elements, including N, Fe, Mg, Ca, Ni, and Ba. We used these data to discuss the evidence of a connection between the distribution of stars along the horizontal branch (HB) and the multiple populations that are typically present in globular clusters. We found that all RHB and most (13 out of 17) BHB stars are O-rich, Na-poor, and N-poor; these stars probably belong to the first stellar generation in this cluster. Only the four warmest observed stars are (moderately) O-poor, Na-rich, and N-rich, and they probably belong to the second generation. While our sample is not fully representative of the whole HB population in NGC 6723, our data suggest that in this cluster only HB stars warmer than ~9000 K, that is one fourth of the total, belong to the second generation, if at all. Since in many other clusters this fraction is about two thirds, we conclude that the fraction of first/second generation in globular clusters may be strongly variable. In addition, the wide range in colour of chemically homogeneous first-generation HB stars requires a considerable spread in mass loss (>0.10 M⊙). The reason for this spread is yet to be understood. Finally, we found a high Ba abundance, with a statistically significant radial abundance gradient. Based on observations collected at ESO telescopes under programme 087.D-0230.Tables 2, 3, 5, 7, and 8 are available in electronic form at http://www.aanda.org