The merger of dark matter halos and the gaseous structures embedded in them, such as galaxies and clusters of galaxies, results in strong shocks. These shocks will produce high-energy neutrinos and gamma-rays through inelastic $pp$ collisions. In this work, we formulate the redshift dependence of the shock velocity, galactic radius, halo gas content and galactic/intergalactic magnetic fields over the dark matter halo distribution up to a redshift $z = 10$. We find that high-redshift mergers contribute a significant amount of the cosmic-ray (CR) luminosity density, and the resulting neutrino spectra could explain a large part of the observed diffuse neutrino flux above 0.1 PeV up to several PeV without violating the $\gamma$-ray constraint. Moreover, the secondary electrons/positrons from pion decays can produce observable emissions through synchrotron radiation and inverse Compton scattering while propagating in the host galaxies. We demonstrate that these emissions can explain the radio and X-ray fluxes of the merging galaxies such as NGC 660 and NGC 3256. In the future, with the improved sensitivity of the Cherenkov Telescope Array (CTA) and IceCube-Gen2, our model for the merging galaxies can be further tested by gamma-ray and neutrino observations.