Populations adapt by acquiring and fixing beneficial mutations. However, the typical number and strength of such mutations remains a subject of debate. In this work, we tackle this question by investigating the genetic basis of long-term adaptation in hybrid yeast populations evolved under different rates of outcrossing. We track the dynamics of adaptation on new and standing variants by resequencing populations and individuals at fixed times over the course of 1000 generations. Additionally, we assay changes in the mean and variance in fitness over time. We find that populations undergoing any amount of sex continue to adapt and maintain substantial variance in fitness through the duration of the experiment, while accumulating few new mutations. Despite this sustained adaptation, allele frequency changes stagnate after several hundred generations, and only a handful of regions that initially experience rapid change proceed to fix. We propose that this phenomenon of sustained adaptation despite stagnating genetic change may be explained by a model of dense, weakly selected sites. In this model, rapid allele frequency changes are the result of many alleles of small effect linked over ten or hundreds of kilobases hitchhiking together. As recombination breaks associations between distant sites, the effect of a typical linkage block declines. However, because individuals at later timepoints sample a greater variety of genotypes, the population continues to adapt. This work suggests that adaptation on weakly selected variants can dominate adaptation for hundreds of generations, and may result in dynamics that may deviate strongly from the selective sweep paradigm.