For each regular coadjoint orbit of a compact group, we construct an exhaustion by symplectic embeddings of toric domains. As a by-product we arrive at a conjectured formula for the Gromov width of coadjoint orbits. Our method combines ideas from Poisson-Lie groups and from the geometric crystals of Berenstein-Kazhdan. We also prove similar results for multiplicity-free spaces. This is joint work with A. Alekseev, J. Lane, and Y. Li.