I am trying to reorder operations on the process chart. Some of them do not move. Are they supposed to?

You can move (or copy and paste) a library operation, and a machining setup along with its operations, on the process chart. Since you cannot move the primary process and its setups and operations--these have a fixed order--move the library operations or machine setups around the primary process. It may be easier to move entries by rolling them up before moves, by clicking the minus sign in the box next to the setup or main process.