  Free groups are known to be homogeneous, meaning that finite tuples of
elements which satisfy the same first-order properties are in the same orbit
under the action of the automorphism group. We show that virtually free groups
have a slightly weaker property, which we call uniform almost-homogeneity: the
set of $k$-tuples which satisfy the same first-order properties as a given
$k$-tuple $\mathbf{u}$ is the union of a finite number of
$\mathrm{Aut}(G)$-orbits, and this number is bounded independently from
$\mathbf{u}$ and $k$. Moreover, we prove that there exists a virtually free
group which is not $\exists$-homogeneous. We also prove that all hyperbolic
groups are homogeneous in a probabilistic sense.
