  We investigate the features of the cosmological expansion history described
by a recent model of gravity characterised by two nonlocally interacting
metrics. We perform a detailed analysis of the dynamical system formed by the
field equations and we find no stable critical points at finite and infinite
distance. Nonetheless, we show that even if the universe does not evolve
towards a de Sitter attractor, the effective equation of state parameter
$\omega_{\rm eff}$ always tends to $-1$, independently from the value of the
free parameter $m^2$, which characterises the nonlocality of the theory. We
also address the behaviour of gravity on Solar System scales and the growth of
small cosmological fluctuations on small scales, in the quasi-static
approximation. We find a post-Newtonian $\gamma$ parameter, a slip parameter
and an effective, normalised gravitational coupling different from unity. These
differences all depend on $m^2$ and are negligible if one consider the
cosmological solution by which $m^2 \sim H_0^2$.
