  Gas kinematics affect the radiative transfer and escape of hydrogen
Lyman-$\alpha$ (Ly$\alpha$) emission from galaxies. We investigate this
interplay empirically by relating the ionised gas kinematics of 42 galaxies in
the extended Ly$\alpha$ Reference Sample with their Ly$\alpha$ escape
fractions, $f_\rm{esc}$, Ly$\alpha$ equivalent widths, $\rm{EW}_\rm{Ly\alpha}$,
and Ly$\alpha$ luminosities, $L_\rm{Ly\alpha}$. To this aim we use PMAS
integral-field spectroscopic observations of the Balmer-$\alpha$ line. We
calculate shearing velocities, $v_\rm{shear}$, and intrinsic velocity
dispersions, $\sigma_0^\rm{obs}$ (empirically corrected for beam-smearing
effects), as global kinematical measures for each galaxy. The sample is
characterised by highly turbulent motions and more than half of the sample
shows dispersion dominated kinematics. We uncover clear trends between
Ly$\alpha$ observables and global kinematical statistics. We discuss
statistically the importance of $v_\rm{shear}$, $\sigma_0^\rm{obs}$, and
$v_\rm{shear}/\sigma_0^\rm{obs}$ for regulating the Ly$\alpha$ observables in
comparison to other galaxy parameters. It emerges that $\sigma_0^\rm{obs}$ is
the dominating parameter for $\rm{EW}_\rm{Ly\alpha}$ and that is as important
as nebular extinction, gas covering fraction, and ionising photon production
efficiency in regulating $f_\rm{esc}$. A simple scenario where the starburst
age is simultaneously regulating turbulence, $\rm{EW}_\rm{Ly\alpha}$, and
$f_\rm{esc}$ does not find support by our observations. However, we show that
the small scale distribution of dust appears to be influenced by turbulence in
some galaxies. In support of our observational result we discuss how turbulence
is theoretically expected to play a significant role in modulating
$f_\rm{esc}$.
  (abridged)
