the.com/running gag

means A joke, phrase, or bit that recurs throughout a show, conversation, or piece of writing, gaining comic power each time it resurfaces.

from A 20th-century theatrical and broadcasting term: 'gag' has meant 'a joke or comic bit' in stage slang since the 1800s, likely from the older sense of 'gag' as something stuffed in the mouth — a comedian's bit was originally ad-libbed material 'gagged' into a script. 'Running' simply marks it as ongoing, carried forward episode after episode. The phrase took hold with vaudeville and radio comedy, where writers leaned on recurring catchphrases and setups the audience could anticipate — the anticipation itself becoming part of the joke.