  In this paper, new results concerning the phenomenon of adiabatic trapping
into resonance for a class of quasi-integrable maps with a time-dependent
exciter are presented and discussed in detail. The applicability of the results
about trapping efficiency for Hamiltonian systems to the maps considered is
proven by using perturbation theory. This allows determining explicit scaling
laws for the trapping properties. These findings represent a generalization of
previous results obtained for the case of quasi-integrable maps with parametric
modulation as well as an extension of the work by Neishtadt \textit{et al.} on
a restricted class of quasi-integrable systems with time-dependent exciters.
