  This paper is devoted to the study of geodesic distances defined on a
subdomain of a given Carnot group, which are bounded both from above and from
below by fixed multiples of the Carnot-Carath\'{e}odory distance. We show that
the uniform convergence (on compact sets) of these distances can be
equivalently characterized in terms of $\Gamma$-convergence of several kinds of
variational problems. Moreover, we investigate the relation between the class
of intrinsic distances, their metric derivatives and the sub-Finsler convex
metrics defined on the horizontal bundle.
