Mid-resolution spectra are used to deduce the size and kinematics of the coronal region in a sample of Seyfert galaxies by means of observations of the [Fe XI], [Fe X], [Fe VII], [Si VI], and [Si VII] lines. These coronal lines (CLs) extend from the unresolved nucleus up to a few tens to a few hundreds of parsecs. The region of the highest ionized ions studied, [Fe XI] and [Fe X], is the least spatially extended and concentrates at the center; intermediate-ionization lines extend from the nucleus up to a few tens to a few hundred parsecs; lower [O III]-like ions are known to extend to the kpc range. All together indicate a stratification in the ionized gas, usually interpreted in terms of nuclear photoionization as the driving ionization mechanism. However, CL profiles show various peculiarities: they are broader by a factor of 2 than lower ionization lines, the broadening being in terms of asymmetric blue wings, and their centroid position at the nucleus is blueshifted by a few hundred km s-1. Moreover, in NGC 1386 and NGC 1068, a double-peaked [Fe VII] line is detected in the nuclear and extended coronal region, this being the first report of this type of profile in CLs in active galactic nuclei. If interpreted as outflow signatures, the total broadening of the lines at zero-intensity levels implies gas velocities up to 2000 km s-1. Although the stratification of ions across the coronal region means that photoionization is the main power mechanism, the high velocities deduced from the profiles, the relatively large spatial extension of the emission, and the results from photoionization models indicate that an additional mechanism is at work. We suggest that shocks generated by the outflow could provide the additional required power for line formation. Based on observations made with ESO Telescopes at the Paranal Observatory under program 68.B-0627.