  We characterize Lipschitz morphisms between quantum compact metric spaces as
those *-morphisms which preserve the domain of certain noncommutative analogues
of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary,
lower semi-continuous Lip-norms with a shared domain are in fact equivalent. We
then note that when a family of lower semi-continuous Lip-norms are uniformly
equivalent, then they give rise to totally bounded classes of quantum compact
metric spaces, and we apply this observation to several examples of
perturbations of quantum metric spaces. We also construct the noncommutative
generalization of the Lipschitz distance between quantum compact metric spaces.
