Kubernetes admission controllers, specifically mutating admission webhooks, were used to ease the process of migrating a staging environment to a new one. The approach involved creating a webhook called "patcher" that intercepts requests to create or modify certain resources and transforms manifests as needed. This allowed for a smooth transition between environments while minimizing disruptions. To ensure reliability and security, the webhook was designed with availability in mind and limited to modifying specific objects. The use of mutating admission controllers introduced some complexities, such as differences between source and live manifests, but ultimately enabled the successful migration of applications to the new cluster.