Feedback mediated by cosmic rays (CRs) is an important process in galaxy formation. Because CRs are long-lived and because they are transported along magnetic field lines independently of any gas flow, they can efficiently distribute their feedback energy within the galaxy. We present an in-depth investigation of (i) how CRs launch galactic winds from a disc that is forming in a $10^{11}{\mathrm{M}}_{\odot }$ halo and (ii) how CR transport affects the dynamics in a galactic outflow. To this end, we use the Arepo moving-mesh code and model CR transport with the two-moment description of CR hydrodynamics. This model includes the CR interaction with gyroresonant Alfv'en waves that enables us to self-consistently calculate the CR diffusion coefficient and CR transport speeds based on coarse-grained models for plasma physical effects. This delivers insight into key questions such as whether the effective CR transport is streaming-like or diffusive-like, how the CR diffusion coefficient and transport speed change inside the circumgalactic medium (CGM), and to what degree the two-moment approximation is needed to faithfully capture these effects. We find that the CR-diffusion coefficient reaches a steady-state in most environments with the notable exception of our newly discovered Alfv'en-wave dark regions where the toroidal wind magnetic field is nearly perpendicular to the CR pressure gradient so that CRs are unable to excite gyroresonant Alfv'en waves. However, CR transport itself cannot reach a steady-state and is not well described by either the CR streaming paradigm, the CR diffusion paradigm or a combination of both.