We consider non-local Integral Kernel Theories of Gravity in a homogeneous
and isotropic universe background as a possible scenario to drive the cosmic
history. In particular, we investigate the cosmological properties of a
gravitational action containing the inverse d'Alembert operator of the Ricci
scalar proposed to improve Einstein's gravity at both high and low-energy
regimes. In particular, the dynamics of a physically motivated non-local
exponential coupling is analyzed in detail by recasting the cosmological
equations as an autonomous system of first-order differential equations with
dimensionless variables. Consequently, we study the phase-space domain and its
critical points, investigating their stability and main properties. In
particular, saddle points and late-time cosmological attractors are discussed
in terms of the free parameters of the model. Finally, we discuss the main
physical consequences of our approach in view of dark energy behavior and the