  We investigate the impact of cosmic rays (CR) and different modes of CR
transport on the properties of Milky Way-mass galaxies in cosmological
magneto-hydrodynamical simulations in the context of the AURIGA project. We
systematically study how advection, anisotropic diffusion and additional
Alfv\'en-wave cooling affect the galactic disc and the circum-galactic medium
(CGM). Global properties such as stellar mass and star formation rate vary
little between simulations with and without various CR transport physics,
whereas structural properties such as disc sizes, CGM densities or temperatures
can be strongly affected. In our simulations, CRs affect the accretion of gas
onto galaxies by modifying the CGM flow structure. This alters the angular
momentum distribution which manifests itself as a difference in stellar and
gaseous disc size. The strength of this effect depends on the CR transport
model: CR advection results in the most compact discs while the Alfv\'en-wave
model resembles more the AURIGA model. The advection and diffusion models
exhibit large ($r\sim50$ kpc) CR pressure-dominated gas haloes causing a
smoother and partly cooler CGM. The additional CR pressure smoothes small-scale
density peaks and compensates for the missing thermal pressure support at lower
CGM temperatures. In contrast, the Alfv\'en-wave model is only CR pressure
dominated at the disc-halo interface and only in this model the gamma-ray
emission from hadronic interactions agrees with observations. In contrast to
previous findings, we conclude that details of CR transport are critical for
accurately predicting the impact of CR feedback on galaxy formation.
