To automate the process of changing a MongoDB deployment's storage engine to WiredTiger, users can utilize MMS Automation, which simplifies the conversion by automatically shutting down secondary mongod processes, emptying their data directories, and starting new instances with the updated configuration file. The automation agent connects to each secondary process in sequence, resynchronizing the replica set once all nodes have completed the conversion, allowing for a seamless transition without downtime or extensive operational work beyond initial setup.