  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 (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 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
\citet{Globus2015b}, 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.
