  We define a new family of discrete representations of relatively hyperbolic
groups which unifies many existing definitions and examples of geometrically
finite behavior in higher rank. The definition includes the relative Anosov
representations defined by Kapovich-Leeb and Zhu, and Zhu-Zimmer, as well as
holonomy representations of various different types of "geometrically finite"
convex projective manifolds. We prove that these representations are all stable
under deformations whose restriction to the peripheral subgroups satisfies a
dynamical condition, in particular allowing for deformations which do not
preserve the conjugacy class of the peripheral subgroups.
