It is shown that solitary-wave, kinklike structures can propagate superluminally in two- and four-level amplifying media with strongly damped oscillations of coherences. This is done by solving analytically the Maxwell-Bloch equations in the kinetic limit. It is also shown that the true wave fronts--unlike the pseudo wave fronts of the kinks--must propagate with velocity c, so that no violation of special relativity is possible. The conditions of experimental verification are discussed.