  In this study, we investigate the growth of structures within the
Deser-Woodard nonlocal theory and extend it to various bouncing cosmology
scenarios. Our findings show that the observable structure growth rate,
$f\sigma_8$, in a vacuum-dominated universe is finite within the redshift range
of $0<z<2$, contrary to previous literature. Although $f\sigma_8$ exhibits no
divergences, we observe a slight difference between the evolution of the
$\Lambda$CDM and the non-local DW II models. Regarding structure formation in
bouncing cosmologies, we evaluate the evolution of $f\sigma_8$ near the
bouncing point. Among the different bouncing cases we explore, the oscillatory
bounce and pre-inflationary asymmetrical bounce demonstrate a physical profile
where the growth rate begins as a small perturbation in the early epoch and
increases with inflation, which can be regarded as the seeds of large-scale
structures. These findings are significant because they shed light on the
growth of seed fluctuations into cosmic structures resulting from non-local
effects.
