One of the most important realizations of the past fifteen years is the vital role that feedback processes must play in the evolution of galaxies, particularly at the massive end (M_star >~ 10^11 Msun). Beginning already within the first 2Gyr of the history of the universe, star formation in massive galaxies appears to be efficiently and rapidly shut off ("quenched") as they transition to the red sequence. I will discuss my ongoing efforts to understand the mechanisms by which galaxies quench, focusing on three populations across redshifts. First, I will show highlights from an ongoing ALMA program to spatially and spectrally resolve outflowing molecular winds from z=4-5 dusty star-forming galaxies. Next, I will show results from a VLA program studying a population of compact galaxies at z~2.5 thought to be rapidly transitioning to quiescence. Finally, I will describe upcoming ALMA observations of z~0.7 galaxies which have already quenched, in an effort to link these objects with local quiescent galaxies. The ultimate goal of these studies is to understand the mechanisms, direction, and consequences of feedback processes in galaxies, and I will outline areas where I expect progress to be made over the next several years.