I'm curious as to how transporting minors across state lines works,
Usually they get into the vehicle, sit down, and then someone drives. from there it's a whirlwind of bathroom breaks,
forgotten meds, motion sickness, and "Johnny's touching my stuff!"
is there a different form that needs to be filled out?
No, at least not from a CAP perspective. You'll have the 32s, so that covers any permission issues.
Some wings will require you get explicit Wing CC approval before leaving the state on official CAP business.
As a CYA, I would suggest you at least 3
seniors (more is better), at least twice as many members as required with CAP driver's
licenses if you are using COVs.
You might also consider making your life considerably less "exciting", make it a "meet you there" activity which can then include families,
and forgo driving cadets.
But from your OP, it sounds like you have what you need to get started, absent anything specific to your wing.