  We identify the first quiescent galaxies in TNG300, the largest volume of the
IllustrisTNG cosmological simulation suite, and explore their quenching
processes and time evolution to z=0. We find that the first quiescent galaxies
with stellar masses M_* > 3 x 10^{10} M_sun and specific star formation rates
sSFR < 10^{-11} yr^{-1} emerge at z~4.2 in TNG300. Suppression of star
formation in these galaxies begins with a thermal mode of AGN feedback at z~6,
and a kinetic feedback mode acts in each galaxy by z~4.7 to complete the
quenching process, which occurs on a time-scale of ~0.35 Gyr. Surprisingly, we
find that the majority of these galaxies are not the main progenitors of their
z=0 descendants; instead, four of the five galaxies fall into more massive
galaxies in subsequent mergers at a range of redshifts 2.5 < z < 0.2. By z=0,
these descendants are the centres of galaxy clusters with average stellar
masses of 8 x 10^{11} M_sun. We make predictions for the first quenched
galaxies to be located by the James Webb Space Telescope (JWST).
