  GeV-TeV gamma-ray 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 (EGB) 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 $\gamma$-rays and neutrinos produced during the
UHECRs propagation and derive constraints on the possible cosmological
evolution of UHECR sources. In particular, we show that the mixed-composition
scenario 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.
