  We investigate the cosmological implications of an effective gravitational
action, inspired by Sakharov's idea of induced gravity, containing non-local
contributions from the operator $\left(\Box +\beta \right)^{-1} R$. The $\beta$
term is a novel feature in the panorama of non-local models of gravity, and
arises naturally within Sakharov theory from the potential of a non-minimally
coupled scalar field after a spontaneous symmetry breaking takes place. In this
class of models the non-local contribution can acquire an oscillatory
behaviour, thereby avoiding the divergence of the Hubble parameter at late
times, which is another characteristic feature of the non-local models treated
in the literature. Furthermore, the effective gravitational coupling $G_{\rm
eff}$ inherits the oscillatory behaviour, resulting in alternating epochs of
stronger and weaker gravity. This framework is argued to have potentially
interesting implications for the $H_0$ and the $\sigma_8$ tensions.
