We have analyzed a large number of Caii H line profiles at the sites of the bright points in the interior of the network using a 35-min-long time sequence of spectra obtained at the Vacuum Tower Telescope (VTT) of the Sacramento Peak Observatory on a quiet regon of the solar disc and studied the dynamical processes associated with these structures. Our analysis shows that the profiles can be grouped into three classes in terms of their evolutionary behaviour. It is surmized that the differences in their behaviour are directly linked with the inner network photospheric magnetic points to which they have been observed to bear a spatial correspondence. The light curves of these bright points give the impression that the ‘main pulse’, which is the upward propagating disturbance carrying energy, throws the medium within the bright point into a resonant mode of oscillation that is seen as the follower pulses. The main pulse as well as the follower pulses have identical periods of intensity oscillations, with a mean value around 190 ± 20 s. We show that the energy transported by these main pulses at the sites of the bright points over the entire visible solar surface can account for a substantial fraction of the radiative loss from the quiet chromosphere, according to current models.