  Small-scale magnetic field concentrations (magnetic elements) in the quiet
Sun are believed to contribute to the energy budget of the upper layers of the
Sun's atmosphere, as they are observed to support a large number of MHD modes.
In recent years, kink waves in magnetic elements were observed at different
heights in the solar atmosphere, from the photosphere to the corona. However,
the propagation of these waves has not been fully evaluated. Our aim is to
investigate the propagation of kink waves in small magnetic elements in the
solar atmosphere. We analysed spectropolarimetric data of high-quality and long
duration of a photospheric quiet Sun region observed near the disk center with
the spectropolarimeter CRISP at the Swedish Solar Telescope (SST), and
complemented by simultaneous and co-spatial broad-band chromospheric
observations of the same region. Our findings reveal a clear upward propagation
of kink waves with frequency above $~2.6$ mHz. Moreover, the signature of a
non-linear propagation process is also observed. By comparing photospheric to
chromospheric power spectra, no signature of an energy dissipation is found at
least at the atmospheric heights at which the data analysed originate. This
implies that most of the energy carried by the kink waves (within the frequency
range under study $< 17$ mHz) flows to upper layers in the Sun's atmosphere.
