  In this paper, we combine separate works on (a) the transfer of infinitesimal
rigidity results from an Euclidean space to the next higher dimension by
coning, (b) the further transfer of these results to spherical space via
associated rigidity matrices, and (c) the prediction of finite motions from
symmetric infinitesimal motions at regular points of the symmetry-derived orbit
rigidity matrix. Each of these techniques is reworked and simplified to apply
across several metrics, including the Minkowskian metric $\M^{d}$ and the
hyperbolic metric $\H^{d}$. This leads to a set of new results transferring
infinitesimal and finite motions associated with corresponding symmetric
frameworks among $\E^{d}$, cones in $E^{d+1}$, $\SS^{d}$, $\M^{d}$, and
$\H^{d}$. We also consider the further extensions associated with the other
Cayley-Klein geometries overlaid on the shared underlying projective geometry.
