  Motivated by practical generalizations of the classic $k$-median and
$k$-means objectives, such as clustering with size constraints, fair
clustering, and Wasserstein barycenter, we introduce a meta-theorem for
designing coresets for constrained-clustering problems. The meta-theorem
reduces the task of coreset construction to one on a bounded number of ring
instances with a much-relaxed additive error. This reduction enables us to
construct coresets using uniform sampling, in contrast to the widely-used
importance sampling, and consequently we can easily handle constrained
objectives. Notably and perhaps surprisingly, this simpler sampling scheme can
yield coresets whose size is independent of $n$, the number of input points.
  Our technique yields smaller coresets, and sometimes the first coresets, for
a large number of constrained clustering problems, including capacitated
clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in
minor-excluded graph, and polygon clustering under Fr\'{e}chet and Hausdorff
distance. Finally, our technique yields also smaller coresets for $1$-median in
low-dimensional Euclidean spaces, specifically of size
$\tilde{O}(\varepsilon^{-1.5})$ in $\mathbb{R}^2$ and
$\tilde{O}(\varepsilon^{-1.6})$ in $\mathbb{R}^3$.
