Tidal dwarf galaxies (TDGs) are recycled objects that form within the collisional debris of interacting/merging galaxies. They are expected to be devoid of non-baryonic dark matter, since they can form only from dissipative material ejected from the discs of the progenitor galaxies. We investigate the gas dynamics in a sample of six bona-fide TDGs around three interacting and post-interacting systems: NGC 4694, NGC 5291, and NGC 7252 ("Atoms for Peace"). For NGC 4694 and NGC 5291 we analyse existing HI data from the Very Large Array (VLA), while for NGC 7252 we present new HI observations from the Jansky VLA together with long-slit and integral-field optical spectroscopy. For all six TDGs, the HI emission can be described by rotating disc models. These HI discs, however, have undergone less than a full rotation since the time of the interaction/merger event, raising the question of whether they are in dynamical equilibrium. Assuming that these discs are in equilibrium, the inferred dynamical masses are consistent with the observed baryonic masses, implying that TDGs are devoid of dark matter. This puts constraints on putative "dark discs" (either baryonic or non-baryonic) in the progenitor galaxies. Moreover, TDGs seem to systematically deviate from the baryonic Tully-Fisher relation. These results provide a challenging test for alternative theories like MOND.