As I do my best to absorb the DP/DEFO rule to the point where it becomes second nature, this thought occurs. My understanding is that the DP can take any defensive position at any time (except DEFO) and not affect the lineup. Any position? Any time? Does that mean that DP can play 1st base for the first batter, sit down for the 2nd and then play left field for the third batter so long as she doesn't take the DEFO spot? Could she theoretically move from position to position with every pitch?

I know that scenario sounds highly improbable but it helps me to take things to their unlikely extreme to fully understand the rules.
And stranger things have happened.

The DP playing for any defensive player other than the DEFO is a defensive position swap, so no players leave the game. The DP will continue to bat in the original position in the order, as will the defensive players the DP is fielding for continue to bat in their original positions.

Just like your starting SS can theoretically move from defensive postion to position with every pitch, so can the DP (theoretically).

The DP can play defense for the DEFO, but in that case the DEFO has left the game. This doesn't affect the batting order, either, but the DEFO will use up her re-entry rights when she comes back in on defense. If the DEFO bats, she must bat for the DP, and in this case, the DP has left the game.