  By making use of Halperin's local systems over simplicial sets and the model
structure of the category of diffeological spaces due to Kihara, we introduce a
framework of rational homotopy theory for such smooth spaces with arbitrary
fundamental groups. As a consequence, we have an equivalence between the
homotopy categories of fibrewise rational diffeological spaces and an algebraic
category of minimal local systems elaborated by G\'omez-Tato, Halperin and
Tanr\'e. In the latter half of this article, a spectral sequence converging to
the singular de Rham cohomology of a diffeological adjunction space is
constructed with the pullback of relevant local systems. In case of a
stratifold obtained by attaching manifolds, the spectral sequence converges to
the Souriau--de Rham cohomology algebra of the diffeological space. By using
the pullback construction, we also discuss a local system model for a
topological homotopy pushout.
