Well, it is about moving attached components that are spawned when the players login, so it is not simple to remake (But I understand that it is hard to find the problem also), so I made a clone of the project, and I cleared out evvverything that is not necessary for the mechanics or the visuals of the problem.
Now all the project has: 1 map, 1 game mode, that finds the decks, and tells them to fill up. 1 DeckBP, which can only spawn attached cards and rearranges them.
See that by default, the client window sees this bug: The cards move to their place slowly (I set it to 2 sec), then at the end every card jumps back to Relative(0,0,0).
If you open the CardBP, and at the MoveTo event you rearrange 2 connections so the card would jump to its place without Timeline and Lerp, then the cards stay where they should be.
I have no idea what could be the problem.