In most battles, it would be advantageous for one party member to start in one spot as opposed to another. Is it possible to swap their start locations or influence it in any way?
From what I can see, you can get a different party layout if you choose 2 rabbids as opposed to 1 mushroom kingdom member and 1 rabbid. It doesn't appear to be a fine-grained control but it can help in some cases.
To clarify Seiyria's answer, the level starting positions are basically fixed and ordered 1-2-3. The starting positions are filled in your team roster order, which is determined by reading the selected characters from left to right in the team selection screen. Mario is on the far left, so is always position one. The next selected character as you read from left to right is number two, and the rightmost one is number three.
With this in mind, you can exert limited control over the starting positions. If you really want, for example, Luigi to start in position #2, then select a party member that sequentially comes after Luigi on the team selection screen to be your third member. If you want Luigi to be in position #3, select a character that comes sequentially before Luigi (i.e. Rabbid Peach or Rabbid Luigi) to be your third member.