Double-check that you have the Sync on on both devices, and that the project is NOT open on both at the same time; Rush handles projects like library books: it can only be "checked out to one person" (i.e., open on one device) at a time.
Since your phone seems to have no issues with the project's assets, make sure that when you do open the project there, the SYNC indicator (blue circular arrow near the Undo/Redo arrow) has disappeared, indicating that the project is now backed up/loaded from the cloud. Close the project on your phone, and then try opening on your iPad. When you do open it there, make sure the same SYNC blue circular arrow completes its rotation before starting to edit/move assets on the timeline.
(I have on occasion still had issues going from iPhone to iPad, and had success when I DUPLICATED a project and then opened the copy on the other device.)
Hope this helps! Make sure to press "✔ Correct Answer" on this post if this answers your question. Happy Creating! Anissa • @anissat