“Do your own research” is used to shut up people who are asking for evidence. It marks the end of any sort of productive discussion. Maybe a good response would be to ask “So you don’t have any evidence?”. If the other party doesn’t respond, they look weak, but if they had evidence, they wouldn’t have tried to pull the “Do your own research” bit. When they reply, presumably without any evidence, proceed to ridicule their lack of evidence.