  We study the discretization of generalized Wasserstein distances with
nonlinear mobilities on the real line via suitable discrete metrics on the cone
of N ordered particles, a setting which naturally appears in the framework of
deterministic particle approximation of partial differential equations. In
particular, we provide a $\Gamma$-convergence result for the associated
discrete metrics as $N \to \infty$ to the continuous one and discuss
applications to the approximation of one-dimensional conservation laws (of
gradient flow type) via the so-called generalized minimizing movements, proving
a convergence result of the schemes at any given discrete time step $\tau>0$.
This the first work of a series aimed at shedding new lights on the interplay
between generalized gradient-flow structures, conservation laws, and
Wasserstein distances with nonlinear mobilities.
