We study the consequences of the $f(R/Box)$ gravity models for the Solar system and the large scale structure of the universe. The spherically symmetric solutions can be used to obtain bounds on the constant and the linear parts of the correction terms. The evolution of cosmological matter structures is shown to be governed by an effective time dependent Newtons constant. We also analyze the propagation of the perturbation modes. Tensor and vector modes are only slightly modified, but two new scalar degrees of freedom are present. Their causality and stability is demonstrated, and their formal ghost conditions are related to a singularity of the cosmological background. In general, the Newtonian limit of these models has no apparent conflicts with observations but can provide useful constraints.
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