  This paper studies the asymptotic behavior of the integral kernel of the
Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed
and the other tends to infinity either within a Weyl chamber of the associated
reflection group, or within a suitable complex domain. The obtained results are
based on the asymptotic analysis of an associated system of ordinary
differential equations. They generalize the well-known asymptotics of the
confluent hypergeometric function $\phantom{}_1F_1$ to the higher-dimensional
setting and include a complete short-time asymptotics for the Dunkl-type heat
kernel. As an application, it is shown that the representing measures of
Dunkl's intertwining operator are generically continuous.
