We describe how various hadronic cascade models, which are implemented in the geant4 toolkit, describe proton and charged pion transverse momentum spectra from p + Cu and Pb collisions at 3, 8, and 15 GeV/c, recently measured in the hadron production (HARP) experiment at CERN. The Binary, ultrarelativistic quantum molecular dynamics (UrQMD) and modified FRITIOF (FTF) hadronic cascade models are chosen for investigation. The first two models are based on limited (Binary) and branched (UrQMD) binary scattering between cascade particles which can be either a baryon or meson, in the three-dimensional space of the nucleus, while the latter (FTF) considers collective interactions between nucleons only, on the plane of impact parameter. It is found that the slow (pT⩽0.3 GeV/c) proton spectra are quite sensitive to the different treatments of cascade pictures, while the fast (pT>0.3 GeV/c) proton spectra are not strongly affected by the differences between the FTF and UrQMD models. It is also shown that the UrQMD and FTF combined with Binary (FTFB) models could reproduce both proton and charged pion spectra from p + Cu and Pb collisions at 3, 8, and 15 GeV/c with the same accuracy