Cosmic rays (CRs) are accelerated in astrophysical systems, such as starburst galaxies, active galactic nuclei and large-scale shocks. These accelerators are usually embedded in the overdense part of the cosmic web, i.e., galaxy clusters and filaments, which are threaded with magnetic fields. Many studies only model the the transport of charged particles in magnetic fields and ignore the secondary neutrons. We explored an alternative route for CR escape—assisted by temporary charge neutral particles, i.e., neutrons. These neutrons are produced in the hadronic interactions of CR protons, and with the aid of time dilation, they may traverse distances larger than galactic scales before decaying into protons. We capture this alternative CR escape channel with a stochastic Poisson term in the transport equation, which also accounts for other processes that attenuate CR flux. We model the CR escape with the neutron-assisted channel for several test cases, representing host sites being a galaxy, galaxy cluster, and filament. We find that the neutron-assisted escape channel is most efficient for ultra-high-energy cosmic rays (E >~ 1e18 eV) and may modify the energy spectrum of CRs at the highest energies. I will present our methodology for modelling CR escape and highlight the importance of incorporating the physics of neutron-assisted CR escape.