Asymmetric inheritance of cellular content through cell division plays an important role in cell viability and fitness. The dynamics of RNA segregation are so far largely unaddressed. This is partly due to a lack of approaches to follow RNAs over multiple cellular divisions. Here, we establish an approach to quantify RNA dynamics in single cells across several generations in a microfluidics device by tagging RNAs with the diSpinach aptamer. Using S. cerevisiae as a model, we quantitatively characterize intracellular RNA transport from mothers into their buds. Our results suggest that, at cytokinesis, ENO2 diSpinach RNA is preferentially distributed to daughters. This asymmetric RNA segregation depends on the lifespan regulator Sir2 and decreases with increasing replicative age of mothers but does not result from increasing cell size during aging. Overall, our approach opens more opportunities to study RNA dynamics and inheritance in live budding yeast at the single-cell level.