So if a monument is completed can it be teleported along with the alliance leader or is it permanent once it's placed or can it be taken down and put back up after?
Monuments can be teleported separately. To teleport a monument:
buy the Monument Teleport item from the Gold Store (only available to alliance leader),
you need another 3x3 tile (doesn't matter who owns the tile, as long as it is from a member of your alliance)
click on any tile within your 3x3 territory, and select teleport. You will be prompted with a screen to select between a City Teleport or a Monument Teleport (only available to leader)
If you do not meet the requirements to teleport the monument, there will be a screen highlighting the requirements you need (in red).