  We study relations between maps between relatively hyperbolic groups/spaces
and quasisymmetric embeddings between their boundaries. More specifically, we
establish a correspondence between (not necessarily coarsely surjective)
quasi-isometric embeddings between relatively hyperbolic groups/spaces that
coarsely respect peripherals, and quasisymmetric embeddings between their
boundaries satisfying suitable conditions. Further, we establish a similar
correspondence regarding maps with at most polynomial distortion.
  We use this to characterise groups which are hyperbolic relative to some
collection of virtually nilpotent subgroups as exactly those groups which admit
an embedding into a truncated real hyperbolic space with at most polynomial
distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
