  We consider the question of continuity of limit sets for sequences of
geometrically finite subgroups of isometry groups of rank-one symmetric spaces,
and prove analogues of classical (Kleinian) theorems in this context. In
particular we show that, assuming strong convergence of the sequence of
subgroups, the limit sets vary continuously with respect to Hausdorff distance,
and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston
maps also converges uniformly to a limiting Cannon-Thurston map. Our approach
uses the theory of extended geometrically finite representations, developed
recently by the second author.
