The aim of the paper is to study qualitative behavior of solutions to a hyperbolic balance law. The main new feature with respect to previous works is that the flux function may have finitely many inflection points, intervals in which it is affine, and corner points. The reaction function is supposed to be zero at 0 and 1, and positive in between. We prove existence of heteroclinic travelling waves connecting the two constant states for opportune choice of speeds. Finally we analyze the large-time behavior of the Riemann problem with values 0 and 1, showing convergence to one of the travelling waves. The speed of the limiting profile is explicitly characterized.