  Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated
with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$
with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the
Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski
spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $\Pi_q$
of positive semidefinite $q\times q$-matrices and show that they appear as
(locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$.
The latter are given by Bessel functions on $\Pi_q$. Moreover, we determine all
positive definite Olshanski spherical functions and discuss related positive
integral representations for matrix Bessel functions. We also extend the
results to the pairs $(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$ which
are related to the Cartan motion groups of non-compact Grassmannians. Here
Dunkl-Bessel functions of type B (for finite $p$) and of type A (for
$p\to\infty$) appear as spherical functions.
