Using the method of generalized characteristics, we study the large-time structure of admissible weak solutions of a scalar conservation law, in one space variable, with a single inflection point, in the presence of a linear source field, under Cauchy data that are periodic and have mean zero. We show that for all times sufficiently large, the solution stays uniformly in a bounded set which is independent of the size of the data. Moreover, when the data are of class C⁰ then, generically, the number of shocks per period attains a finite value which remains constant thereafter. In particular, all shocks become left-contact discontinuities while their evolution is governed by a nonlinear system of functional and delay-differential equations.