Associate adjunct professor to Assistant adjunct professor in USA
This is likely much more common than for non-adjuncts. The protections for adjuncts are pretty weak generally. However, there are situations in which adjuncts are unionized and then the union rules would apply. 

Also, the typical adjunct position is a fixed term contract that might be renewed or not. The nominal rank would be part of the contract. 

But possible? Certainly. Not common, though.