The observational data on ultrahigh energy cosmic rays (UHECR), in particular their mass composition, show strong indications for extremely hard spectra of individual mass groups of CR nuclei at Earth. It has been shown that such hard spectra can be the result of the finite life-time of UHECR sources, if only a few individual sources dominate the UHECR flux at the highest energies. In this work we investigate the requirements on their CR power and life-time dependent on the characteristics of a purely turbulent, extragalactic magnetic field (EGMF). Without accounting for the anisotropy data, we are able to draw some robust constraints on the contribution of the brightest local radio galaxies, if the initial source spectrum at the acceleration side is soft, i.e. with a spectral index $\alpha\gtrsim 2$, as expected from conventional shock-acceleration theory. Moreover, we show that a local source could provide a dominant UHECR contribution which agrees with the mass composition at energies $E\gtrsim 40$ EeV data only in the case of a strong EGMF.