The nature of cosmic rays (CRs) and their transport in galaxy clusters is probed by several observations. Radio observations reveal synchrotron radiation of cosmic-ray electrons (CRe) spiralling around cluster magnetic fields. γ-ray observations reveal hadronic reactions of cosmic-ray protons (CRp) with gas nuclei that produce pions. No such cluster-wide γ-ray signal has been measured, putting an upper limit on the density of CRp in clusters. But the presence of CRe implies some source of CRp, and consequently there must be some CRp-loss mechanism. We quantify the observational constraints on this mechanism assuming that losses are dominated by CR transport, ultimately deriving lower limits on this transport. Using the Coma cluster as an example, we find that bulk outward speeds of 10–100 km s⁻¹ are sufficient to reduce γ-radiation below current upper limits. These speeds are sub-Alfvénic and are consistent with a self-confinement model for CR transport if the magnetic field is coherent on large scales. If the transport is diffusive, we require minimum diffusion coefficients of 10³¹–10³² cm² s⁻¹. This is consistent with CRs free streaming at the speed of light along a field tangled on length-scales of a few kpc. We find that a model of the Coma cluster with a tangled field and the self-confinement picture together can be consistent with observations if the relative acceleration efficiency of CR protons is less than 15 times more than that of electrons of the same energy. This value is 3–6 times lower than the same quantity for Galactic cosmic rays.