The process by which a counterfeit quotation or news item eventually gets picked up by multiple news sources, which cite one another but never lead back to a verifiable source (because there isn't one). The quote's appearance on multiple forums and news aggregators lends it credibility because it is viewed as "reported" by multiple outlets.
Laundered quotes can originate from satire that is not recognized as satire, but more frequently from an Internet writer who is paraphrasing or outright fabricating.
No matter how many places you see it mentioned, the starlet never actually Tweeted that "Scientology is an abusive cult." A comedian said this while doing an impression of her, and then quote laundering took over.