  This paper deals with the study of parameter dependence of extensions of
Lipschitz mappings from the point of view of continuity. We show that if
assuming appropriate curvature bounds for the spaces, the multivalued extension
operators that assign to every nonexpansive (resp. Lipschitz) mapping all its
nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz
constant) are lower semi-continuous and admit continuous selections. Moreover,
we prove that Lipschitz mappings can be extended continuously even when
imposing the condition that the image of the extension belongs to the closure
of the convex hull of the image of the original mapping. When the target space
is hyperconvex one can obtain in fact nonexpansivity.
