  The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of
probability distributions that enjoys several serendipitous properties. It is
complete separable if the underlying metric space of points is complete
separable, and in that case it metrizes the weak topology. We introduce a
variant of that construction in the realm of quasi-metric spaces, and prove
that it is algebraic Yoneda-complete as soon as the underlying quasi-metric
space of points is algebraic Yoneda-complete, and that the associated topology
is the weak topology. We do this not only for probability distributions,
represented as normalized continuous valuations, but also for subprobability
distributions, for various hyperspaces, and in general for different brands of
functionals. Those functionals model probabilistic choice, angelic and demonic
non-deterministic choice, and their combinations. The mathematics needed for
those results are more demanding than in the simpler case of metric spaces. To
obtain our results, we prove a few other results that have independent
interest, notably: continuous Yoneda-complete spaces are consonant; on a
continuous Yoneda-complete space, the Scott topology on the space of
$\overline{\mathbb{R}}_+$-valued lower semicontinuous maps coincides with the
compact-open and Isbell topologies, and the subspace topology on spaces of
$\alpha$-Lipschitz continuous maps also coincides with the topology of
pointwise convergence, and is stably compact; we introduce and study the
so-called Lipschitz-regular quasi-metric spaces, and we show that the formal
ball functor induces a Kock-Z\"oberlein monad, of which all algebras are
Lipschitz-regular; and we prove a minimax theorem where one of the spaces is
not compact Hausdorff, but merely compact.
