This is true once your passport has a valid vis even if it is expired you can brink it along with a valid pssport.As per my experience in Bangladesh that USA always issue 5 years multiple visa and passport in BD is valid maximum 5 years . Even though passport is expired but visa is valid and good to use subject to you get another new valid passport and travel with both. In Bangladesh I used to carry 6 books of old passport along with one valid . You don’t need to transfer or apply for new visa.
however in our cases canada does not allow you to have Two RTD's at the same time so they will not issue a new one until they have the old one back. so you kinda have no other choise but to get a new US visa at which point the VO will cancel the old visa and provide a new one upon approval and most times you are approved once you tell them the reason and they see that you already have a ten year validity in the old one.
Good Luck.