  We investigate the behavior of the asymptotic late-times effective equation
of state for a class of nonlocal theories of gravity. These theories modify the
Einstein-Hilbert Lagrangian introducing terms containing negative powers of the
d'Alembert operator acting on the Ricci scalar. We find that imposing vanishing
initial conditions for the nonlocal content during the radiation-dominated
epoch implies the same asymptotic late-times behavior for most of these models.
In terms of the effective equation of state of the universe, we find that
asymptotically $\omega_{ \rm eff} \rightarrow -1$, approaching the value given
by a cosmological constant. On the other hand, unlike in the case of
$\Lambda$CDM, the Hubble factor is a monotonic growing function that diverges
asymptotically. We argue that this behavior is not a coincidence and discuss
under which conditions this is to be expected in these nonlocal models.
