The formation of the ‘bump’ and the ‘black-body cutoff’ in the cosmic-ray (CR) spectrum arising from the π-meson photoproduction reaction in collisions of UHE CR protons with the microwave background radiation (MBR) is studied. A kinetic equation which describes CR proton propagation in the MBR with account of the catastrophic nature of the π-meson photoproduction process is derived. The equilibrium CR proton spectrum obtained from the solution of the stationary kinetic equation is in general agreement with the spectrum obtained under assumption of the continuous energy loss approximation. However, the spectra from point sources noticeably differ from those obtained in the continuous loss approximation. Both, the equilibrium and the point source spectra are modified when taking into account the possible deviation of the MBR spectrum from the Planckian one in the Wien region. Thus, for the recently measured MBR spectrum, which reveals an essential ‘excess’ in the submillimeter region, the ‘black-body cutoff’ and the preceding ‘bump’ shift towards lower energies.