If you run out of pieces to put down i.e cultists, warriors, etc. during the summoning phase - what happens? Is the only way to move characters around is from moving them from one area to another?

Your figures are limiting, that is why there is a different mix for each chaos power - to reflect their requirents and style.

Once they are all on the board you have to move them around the board - they can fly a fair distance if you have people scattered about the board