Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II

We consider a continuous solution $u$ of the balance law $${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x(f(u))=g$$ in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term.