  Stellar activity is ubiquitously encountered in M dwarfs and often
characterised by the H$\alpha$ line. In the most active M dwarfs, H$\alpha$ is
found in emission, sometimes with a complex line profile. Previous studies have
reported extended wings and asymmetries in the H$\alpha$ line during flares. We
used a total of 473 high-resolution spectra of 28 active M dwarfs obtained by
the CARMENES (Calar Alto high-Resolution search for M dwarfs with Exo-earths
with Near-infrared and optical Echelle Spectrographs) spectrograph to study the
occurrence of broadened and asymmetric H$\alpha$ line profiles and their
association with flares, and examine possible physical explanations. We
detected a total of 41 flares and 67 broad, potentially asymmetric, wings in
H$\alpha$. The broadened H$\alpha$ lines display a variety of profiles with
symmetric cases and both red and blue asymmetries. Although some of these line
profiles are found during flares, the majority are at least not obviously
associated with flaring. We propose a mechanism similar to coronal rain or
chromospheric downward condensations as a cause for the observed red
asymmetries; the symmetric cases may also be caused by Stark broadening. We
suggest that blue asymmetries are associated with rising material, and our
results are consistent with a prevalence of blue asymmetries during the flare
onset. Besides the H$\alpha$ asymmetries, we find some cases of additional line
asymmetries in \ion{He}{i} D$_{3}$, \ion{Na}{i}~D lines, and the \ion{He}{i}
line at 10830\,\AA\, taken all simultaneously thanks to the large wavelength
coverage of CARMENES. Our study shows that asymmetric H$\alpha$ lines are a
rather common phenomenon in M~dwarfs and need to be studied in more detail to
obtain a better understanding of the atmospheric dynamics in these objects.
