If you move a beacon, you need to delete it and remap it, or move its pin icon to the new location.
To delete and remap - Delete the beacons from the references side bar. To delete all references, click on References -> Delete -> Delete References (All) or Delete References (View) to delete all in the current view. To delete one reference, right click on the reference -> remove location -> right click -> delete reference. Last, run Neon Mapper to remap beacons in their new locations.
To move the beacon's pin icon - click on the beacon to bring up the name tag. Click and drag the name tag to the new location. The pin's location will update when you release the mouse button.