  We investigate on the nonlinear dynamics of Airy beams in a regime where
nonlinear losses due to multi-photon absorption are significant. We identify
the nonlinear Airy beam (NAB) that preserves the amplitude of the inward
H\"ankel component as an attractor of the dynamics. This attractor governs also
the dynamics of finite-power (apodized) Airy beams, irrespective of the
location of the entrance plane in the medium with respect to the Airy waist
plane. A soft (linear) input long before the waist, however, strongly speeds up
NAB formation and its persistence as a quasi-stationary beam in comparison to
an abrupt input at the Airy waist plane, and promotes the formation of a new
type of highly dissipative, fully nonlinear Airy beam not described so far.
