Context. The kinematics of planetary nebulae in external galaxies and in our own is a clue for understanding the behavior of the low- and intermediate-mass stars and their relation with other components of the galaxies. Aims: By measuring precise radial velocities of planetary nebulae (which belong to the intermediate-age population), H ii regions and A-type supergiant stars (which are members of the young population) in NGC 6822, we aim to determine whether both types of population share the kinematics of the disk of H i found in this galaxy. Methods: Spectroscopic data for six planetary nebulae were obtained with the high spectral-resolution spectrograph Magellan Inamori Kyocera Echelle (MIKE) on the Magellan telescope at Las Campanas Observatory. Data for another three PNe and one H ii region were obtained from the SPM Catalog of Extragalactic Planetary Nebulae, which employed the Manchester Echelle Spectrometer attached to the 2.1m telescope at the Observatorio Astronómico Nacional, México. An additional PN and one H ii region were observed with this same telescope-spectrograph in 2013. Thus, in total we have high-quality data for 10 of the 26 PNe detected in this galaxy. In the wavelength calibrated spectra, the heliocentric radial velocities were measured with a precision better than 5-6 km s-1. Data for two additional H ii regions and two A-type supergiant stars were collected from the literature. The heliocentric radial velocities of the different objects were compared to the velocities of the H i disk at the same position. Results: From the analysis of radial velocities we found that H ii regions and A-type supergiants do share the kinematics of the H i disk at the same position, as expected for these young objects. In contrast, most planetary nebula velocities differ significantly (more than 12 km s-1) from that of the H i at the same position. The kinematics of planetary nebulae is different from the young population kinematics and is more similar to the behavior shown by carbon stars, which are intermediate-age members of the stellar spheroid existing in this galaxy. Our results confirm that there are at least two very different kinematical systems in NGC 6822. Based on data obtained at Las Campanas Observatory, Carnegie Institution, Chile.Based on data collected at the Observatorio Astronómico Nacional, SPM. B.C., México.