I do think it's the Lord's decision, not the Duke's.
When the Duke (Duchy A) becomes Lord of a region in another Duchy (B), it will become part of his own Duchy (A) though, I think. At least I've had the case that when being lord in duchy B, then being appointed as duke of duchy A, the region obviously joined duchy A as well.
Is this at all clear?
Any confirmation on that might be needed though. Allegiances have changed a lot, when I was gone and while I was already back. So I could have missed something.