Fast radio bursts (FRBs) are millisecond transients of unknown origin(s) occurring at cosmological distances. Here we, for the first time, show time-integrated-luminosity functions and volumetric occurrence rates of non-repeating and repeating FRBs against redshift. The time-integrated-luminosity functions of non-repeating FRBs do not show any significant redshift evolution. The volumetric occurrence rates are almost constant during the past $sim$10 Gyr. The nearly-constant rate is consistent with a flat trend of cosmic stellar-mass density traced by old stellar populations. Our findings indicate that the occurrence rate of non-repeating FRBs follows the stellar-mass evolution of long-living objects with $sim$Gyr time scales, favouring e.g. white dwarfs, neutron stars, and black holes, as likely progenitors of non-repeating FRBs. In contrast, the occurrence rates of repeating FRBs may increase towards higher redshifts in a similar way to the cosmic star formation-rate density or black hole accretion-rate density if the slope of their luminosity function does not evolve with redshift. Short-living objects with $lesssim$ Myr time scales associated with young stellar populations (or their remnants, e.g., supernova remnants, young pulsars, and magnetars) or active galactic nuclei might be favoured as progenitor candidates of repeating FRBs.
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