I think the first and primary issue will be getting a boarding pass, once you're on the plane you should be fine and will be able to explain to CBP. Flights to the USA get advance passenger processing, every checkin goes to the US for verification and validation of valid visa, or ESTA. If you don't have an ESTA in the passport your travelling in, and your visa is in another passport you will get denied automatic checkin and have to show your documents to the checkin agent. It's common because State Department get you to carry on using a visa even if it is in an expired passport. Other than that though, they should check you in and board you and then you'll be fine. So in summary, it might come down to convincing the checkin agent and auto checkins won't work anymore.