  We prove distributional convergence for a family of random processes on
$\mathbb{Z}$, which we call cooperative motions. The model generalizes the
"totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns,
Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on
connecting a temporal recurrence relation satisfied by the cumulative
distribution functions of the process to the theory of finite difference
schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point
out some surprising lattice effects that can persist in the distributional
limit, and propose several generalizations and directions for future research.
