  A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$
whose vertices are constrained to lie on concentric $d$-spheres with
independently variable radii. In particular, combinatorial characterisations
are established for the rigidity of generic frameworks for $d=1$ with an
arbitrary number of independently variable radii, and for $d=2$ with at most
two variable radii. This includes a characterisation of the rigidity or
flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$.
Due to the equivalence of the generic rigidity between Euclidean space and
spherical space, these results interpolate between rigidity in 1D and 2D and to
some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the
detection of symmetry-induced continuous flexibility in frameworks on spheres
with variable radii are also provided.
