  We study nonlocal conservation laws with a discontinuous flux function of
regularity $\mathsf{L}^{\infty}(\mathbb{R})$ in the spatial variable and show
existence and uniqueness of weak solutions in
$\mathsf{C}\big([0,T];\mathsf{L}^{1}_{\text{loc}}(\mathbb{R})\big)$, as well as
related maximum principles. We achieve this well-posedness by a proper
reformulation in terms of a fixed-point problem. This fixed-point problem
itself necessitates the study of existence, uniqueness and stability of a class
of discontinuous ordinary differential equations. On the ODE level, we compare
the solution type defined here with the well-known Carath\'eodory and Filippov
solutions.
