Recent progress in reconstructing gene regulatory networks has established a framework for a quantitative description of the dynamics of many important cellular processes. Such a description will require novel experimental techniques that enable the generation of time-series data for the governing regulatory proteins in a large number of individual living cells. Here, we utilize microfabrication to construct a Tesla microchemostat that permits single-cell fluorescence imaging of gene expression over many cellular generations. The device is used to capture and constrain asymmetrically dividing or motile cells within a trapping region and to deliver nutrients and regulate the cellular population within this region. We illustrate the operation of the microchemostat with Saccharomyces cerevisiae and explore the evolution of single-cell gene expression and cycle time as a function of generation. Our findings highlight the importance of novel assays for quantifying the dynamics of gene expression and cellular growth, and establish a methodology for exploring the effects of gene expression on long-term processes such as cellular aging.