Not move exactly, but you can "duplicate" your last current ship and multi-tool between your saves going into and coming out of the expedition. No freighters though. So just make sure that the ship and MT you want to have access to "on the other side" is the one that is currently selected before you go to the expedition terminal. They will have all the tech, but your ship's cargo won't be duplicated. You can move cargo though in a different part of the terminal.