Rearrange mission badges
There is no easy way. However, if you redo a mission the badge goes up as the last mission done. So you can redo the whole mosaic missions in the right order.
This also mean you must avoid redoing a mission belonging to such a picture.
There is currently no way to rearrange the order of the icons manually. Each mission icon is displayed in the order in which they were completed with oldest ones at the bottom right – and you can only get credit for a mission once.
You can, however, complete the missions again and the newest icons will reappear at the top.