The heating of the solar chromosphere remains one of the most important questions in solar physics. Our current understanding is that small-scale internetwork (IN) magnetic fields play an important role as a heating agent. Indeed, cancellations of IN magnetic elements in the photosphere can produce transient brightenings in the chromosphere and transition region. These bright structures might be the signature of energy release and plasma heating, probably driven by the magnetic reconnection of IN field lines. Although single events are not expected to release large amounts of energy, their global contribution to the chromosphere may be significant due to their ubiquitous presence in quiet Sun regions. In this paper, we study cancellations of IN elements and analyze their impact on the energetics and dynamics of the quiet Sun atmosphere. We use high-resolution, multiwavelength, coordinated observations obtained with the Interface Region Imaging Spectrograph and the Swedish 1 m Solar Telescope (SST) to identify cancellations of IN magnetic flux patches and follow their evolution. We find that, on average, these events live for similar to 3. minutes in the photosphere and similar to 12. minutes in the chromosphere and/or transition region. Employing multi-line inversions of the Mg II h and k lines, we show that cancellations produce clear signatures of heating in the upper atmospheric layers. However, at the resolution and sensitivity accessible to the SST, their number density still seems to be one order of magnitude too low to explain the global chromospheric heating.