GeV–TeV gamma-rays and PeV–EeV neutrino backgrounds provide a unique window on the nature of the ultra-high-energy cosmic rays (UHECRs). We discuss the implications of the recent Fermi-LAT data regarding the extragalactic gamma-ray background and related estimates of the contribution of point sources as well as IceCube neutrino data on the origin of the UHECRs. We calculate the diffuse flux of cosmogenic γ-rays and neutrinos produced by the UHECRs and derive constraints on the possible cosmological evolution of UHECR sources. In particular, we show that the mixed-composition scenario considered in Globus et al., which is in agreement with both (i) Auger measurements of the energy spectrum and composition up to the highest energies and (ii) the ankle-like feature in the light component detected by KASCADE-Grande, is compatible with both the Fermi-LAT measurements and with current IceCube limits. We also discuss the possibility for future experiments to detect associated cosmogenic neutrinos and further constrain the UHECR models, including possible subdominant UHECR proton sources.