  We study a family $C_{s,l}$ of Capelli-type invariant differential operators
on the space of rectangular matrices over a real division algebra. The
$C_{s,l}$ descend to invariant differential operators on the corresponding
Grassmannian, which is a compact symmetric space, and we determine the image of
the $C_{s,l}$ under the Harish-Chandra homomorphism. We also obtain analogous
results for corresponding operators on the non-compact duals of the
Grassmannians, and for line bundles. As an application we obtain a Radon
inversion formula, which generalizes a recent result of B. Rubin for real
Grassmannians.
