  We construct some classes of dynamical $r$-matrices over a nonabelian base,
and quantize some of them by constructing dynamical (pseudo)twists in the sense
of Xu. This way, we obtain quantizations of $r$-matrices obtained in earlier
work of the second author with Schiffmann and Varchenko. A part of our
construction may be viewed as a generalization of the Donin-Mudrov nonabelian
fusion construction. We apply these results to the construction of equivariant
star-products on Poisson homogeneous spaces, which include some homogeneous
spaces introduced by De Concini.
