"Telephone Game" is the phrase I've seen used to describe this phenomenon.

It may be an instance of the children game called;

• Chinese whispers (or telephone in the United States) is an internationally popular game, in which one person whispers a message to another and so on through a line of people until the last player announces the message to the entire group.

• Although the objective is to pass around the message without it becoming misheard and altered along the way, part of the fun is that, regardless, this usually ends up happening.

Wikipedia