Many astrophysical bodies harbour magnetic fields that are thought to be sustained by a dynamo process. However, it has been argued that the production of large-scale magnetic fields by mean-field dynamo action is strongly suppressed at large magnetic Reynolds numbers owing to the conservation of magnetic helicity. This phenomenon is known as catastrophic quenching. Advection of magnetic fields by stellar and galactic winds towards the outer boundaries and away from the dynamo is expected to alleviate such quenching. Here we explore the relative roles played by advective and turbulent-diffusive fluxes of magnetic helicity in the dynamo. In particular, we study how the dynamo is affected by advection. We do this by performing direct numerical simulations of a turbulent dynamo of alpha(2) type driven by forced turbulence in a Cartesian domain in the presence of a flow away from the equator where helicity changes sign. Our results indicate that in the presence of advection, the dynamo, otherwise stationary, becomes oscillatory. We confirm an earlier result for turbulent-diffusive magnetic helicity fluxes that for small magnetic Reynolds numbers (Rm less than or similar to 100 ... 200, based on the wavenumber of the energy-carrying eddies) the magnetic helicity flux scales less strongly with magnetic Reynolds number (Rm(-1/2)) than the term describing magnetic helicity destruction by resistivity (Rm(-1)). Our new results now suggest that for larger Rm the former becomes approximately independent of Rm, while the latter falls off more slowly. We show for the first time that both for weak and stronger winds, the magnetic helicity flux term becomes comparable to the resistive term for Rm greater than or similar to 1000, which is necessary for alleviating catastrophic quenching.