  Both cosmological expansion and black holes are ubiquitous features of our
observable Universe, yet exact solutions connecting the two have remained
elusive. To this end, we study self-gravitating classical fields within
dynamical spherically symmetric solutions that can describe black holes in an
expanding universe. After attempting a perturbative approach of a known
black-hole solution with scalar hair, we show by exact methods that the unique
scalar field action with first-order derivatives that can source shear-free
expansion around a black hole requires noncanonical kinetic terms. The
resulting action is an incompressible limit of k-essence, otherwise known as
the cuscuton theory, and the spacetime it describes is the McVittie metric. We
further show that this solution is an exact solution to the vacuum
Ho\v{r}ava-Lifshitz gravity with anisotropic Weyl symmetry.
