  In this paper we prove well-posedness for a measure-valued continuity
equation with solution-dependent velocity and flux boundary conditions, posed
on a bounded one-dimensional domain. We generalize the results of [Evers, Hille
and Muntean. Journal of Differential Equations, 259:1068-1097, 2015] to
settings where the dynamics are driven by interactions. In a forward-Euler-like
approach, we construct a time-discretized version of the original problem and
employ the results of [Evers, Hille and Muntean. Journal of Differential
Equations, 259:1068-1097, 2015] as a building block within each subinterval. A
limit solution is obtained as the mesh size of the time discretization goes to
zero. Moreover, the limit is independent of the specific way of partitioning
the time interval $[0,T]$. This paper is partially based on results presented
in Chapter 5 of [Evers. PhD thesis, Eindhoven University of Technology, 2015],
while a number of issues that were still open there, are now resolved.
