When we moved to our new place, the former owners didn't file their new address . . . so, we kept getting their mail. After a few weeks, & they kept saying they'd filed for it, I simply told them that I was going to start sending their mail right back to the senders, instead of calling them to come get it. . . . We never got another piece of their mail again.
A year or so later Police came looking for the man.
He'd gone to jail for abusing his wife, & had escaped.
They told me to keep my doors locked!
Since you know where your tenant lives, you can probably file a Change of Address for them.
It would be better if they do it them-self, though.
Maybe if you tell them what I told the people we bought our place from, they'll go ahead & file it for them-self.
You could at least try that first. And if it doesn't motivate them enough, just start returning their mail to its senders.