@Hermanater1
Yes, Premium will transfer as well. However, you will have to manually download them, they will not download automatically. Make sure to do this via the In-Game store, not XBL Store as XBL Store will try to charge you, and the in-Game store will not. You will have to download the Premium file (very small, a few MB), then you can go back and download the rest of the DLC. Make sure to grab the 3 free DLC from the XBL Store as well (Night, Legacy, and Community Operations).
@lordeleo @Pyramat
If you have not played yet, try the following steps to see if it triggers the prompt to appear:
1. Close the BF4 application on your Xbone or PS4
2. Log into your Battlelog profile on a PC or Laptop
3. Hover over your name in the top-right corner and click "Change Soldier"
4. Under the far left column labeled "Active", hit the one next to your Xbone or PS4 soldier so it turns green
5. Hit "Save"
6. Start BF4 back up on your Xbone or PS4
7. You might now see the prompt