Aims: We aim to improve our picture of the low chromosphere in the quiet-Sun internetwork by investigating the intensity, horizontal velocity, size and lifetime variations of small bright points (BPs; diameter smaller than 0.3 arcsec) observed in the Ca ii H 3968 Å passband along with their magnetic field parameters, derived from photospheric magnetograms. Methods: Several high-quality time series of disc-centre, quiet-Sun observations from the Sunrise balloon-borne solar telescope, with spatial resolution of around 100 km on the solar surface, have been analysed to study the dynamics of BPs observed in the Ca ii H passband and their dependence on the photospheric vector magnetogram signal. Results: Parameters such as horizontal velocity, diameter, intensity and lifetime histograms of the isolated internetwork and magnetic Ca ii H BPs were determined. Mean values were found to be 2.2 km s-1, 0.2 arcsec (≈150 km), 1.48 ⟨ ICa ⟩ and 673 s, respectively. Interestingly, the brightness and the horizontal velocity of BPs are anti-correlated. Large excursions (pulses) in horizontal velocity, up to 15 km s-1, are present in the trajectories of most BPs. These could excite kink waves travelling into the chromosphere and possibly the corona, which we estimate to carry an energy flux of 310 W m-2, sufficient to heat the upper layers, although only marginally. Conclusions: The stable observing conditions of Sunrise and our technique for identifying and tracking BPs have allowed us to determine reliable parameters of these features in the internetwork. Thus we find, e.g., that they are considerably longer lived than previously thought. The large velocities are also reliable, and may excite kink waves. Although these wave are (marginally) energetic enough to heat the quiet corona, we expect a large additional contribution from larger magnetic elements populating the network and partly also the internetwork.