We study the asymptotic behaviour of both spherical $t$-designs and random uniform designs as the set of sampling points in non-parametric regression with spherical regressors of arbitrary dimension. We show that the corresponding regression experiments are asymptotically equivalent, in the sense of Le Cam, to the same sequence of Gaussian white noise experiments as the sample size tends to infinity. More precisely, global asymptotic equivalence is established over spherical Sobolev balls (for both the fixed and the random uniform design case) and over spherical Besov balls (for the fixed design case). Matching non-equivalence results demonstrate that the imposed smoothness assumptions are essentially sharp.