Halo assembly bias is the secondary dependence of the clustering of dark matter haloes on their assembly histories at fixed halo mass. This established dependence is expected to manifest itself on galaxy clustering, a potential effect commonly known as galaxy assembly bias. Using the IllustrisTNG300 magnetohydrodynamical simulation, we analyse the dependence of the properties and clustering of galaxies on the specific mass accretion history of their hosting haloes (sMAH). We first show that several halo and galaxy properties strongly correlate with the slope of the sMAH (beta) at fixed halo mass. Haloes with increasingly steeper beta increment their masses faster early on, and their hosted galaxies present larger stellar-to-halo mass ratios, lose their gas faster, reach the peak of their star formation histories at higher redshift, and become quenched earlier. We also demonstrate that beta provides a more stable link to these key galaxy formation properties than other broadly employed halo proxies, such as formation time. Finally, we measure the secondary dependence of galaxy clustering on beta at fixed halo mass. By tracing back the evolution of individual haloes, we show that the amplitude of the galaxy assembly bias signal for the progenitors of z = 0 galaxies increases with redshift, reaching a factor of 2 at z = 1 for haloes of M-halo = 10(11.)(5) -10(12) h(-1) M-circle dot. The measurement of the evolution of assembly bias along the merger tree provides a new theoretical perspective to the study of secondary bias. Our findings have also important implications for the generation of mock catalogues for upcoming cosmological surveys.