I was experiencing the same problem, that a prop that was getting moved with the MoveTo() method was not updating for clients that were too far away and now found a solution:
In another post, someone pointed out that Cinematic Sequence Devices have a option to make them always relevant, called Level Sequence Actor Always Relevant. So if you parent the prop that you want to move to that sequence device and use the MoveTo() method on the sequence device instead of the prop, it should also update the props position, even from far away. You don’t have to play any sequence on the device, it just acts as parent dummy. For me this worked flawlessly.