  The decomposition of polynomial spaces on unions of Grassmannians $\mathcal
G_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ into irreducible orthogonally
invariant subspaces and their reproducing kernels are investigated. We also
generalize the concepts of cubature points and $t$-designs from single
Grassmannians to unions. We derive their characterization as minimizers of a
suitable energy potential to enable $t$-design constructions by numerical
optimization. We also present new analytic families of $t$-designs for
$t=1,2,3$.
