If you had no warrant and no charges were filed and you moved, and 4 to 7 years go by, can they still make a case of it?

Statute of limitations for misdemeanors is 1 year from the alleged incident. For most felonies, the statute of limitations is 3 years. So if the court never issued a warrant and no charges were filed, after 4-7 years, it's highly unlikely that the prosecutor can charge you with any offense. You can hire an attorney to look into it for you to confirm that no summons was sent to you at a previous address. (It's not the court's responsibility to make sure you receive a summons if you have moved and you didn't update your address.)