  Given a complete non-compact Riemannian manifold $(M,g)$ with certain
curvature restrictions, we introduce an expansion condition concerning a group
of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the
sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these
conditions, compact Sobolev-type embeddings \`a la Berestycki-Lions are proved
for the full range of admissible parameters (Sobolev, Moser-Trudinger and
Morrey). We also consider the case of non-compact Randers-type Finsler
manifolds with finite reversibility constant inheriting similar embedding
properties as their Riemannian companions; sharpness of such constructions are
shown by means of the Funk model. As an application, a quasilinear PDE on
Randers spaces is studied by using the above compact embeddings and variational
arguments.
