We use the framework of Morse theory with differential graded coefficients to study certain operations on the total space of a fibration. More particularly, we focus in this paper on a chain-level description of the Chas-Sullivan product on the homology of the free loop space of an oriented, closed and connected manifold. The idea of "intersecting on the base" and "concatenating on the fiber" are well-adapted to this framework. We also give a Morse theoretical description of other products that follow the same principle. For this purpose, we develop functorial properties with respect to the coefficient in terms of morphisms of A ∞ -modules and morphisms of fibrations. We also build a differential graded version of the Künneth formula and of the Pontryagin-Thom construction.