Client parts that move along with the player using AlignPosition

Due to some compatibility reasons with my game, I need to make the moving parts that move along with the player client sided. Keep in mind that I don’t want to weld the player to the part since I still want the player to be able to freely move on the part while the moving part itself is moving.

This is how the moving part behaves when the RunContext of the script is set to Client. The only issue with this is that the player does not move along with the part.
Client

This is how the moving part behaves when the RunContext of the script is set to Server. I want the same functionality but on the client.
Server