Pro drivers have a lot of experience flying to races. They know not to have certain things in their stuff when they go to the airport. Aerosol cans, bottles of unmarked petrochemicals (LW in an old sauce bottle), ect...

They probably go to the store when they get in town and stock up on stuff just like normal people. Or, they have arrangements made to have local team mates make sure they have stuff like that for them.

When I flew to the CRCRC race, one of the guys from my group had his sponsor ship a pre-production stadium truck kit to our hotel. That truck put his baggage over weight, and we had all sorts of fun because of the musical checked luggage. Long short of it was that my friend and I ended up at the TSA desk over a film canister of baking soda in my stuff, which my friend had checked in his name. After an explanation of what it was and how it was used, they kept the baking soda (only because they asked, and I said "okay"), and let us go.

Travelling to races can be a logistical nightmare, but you'd be surprised how good those guys are at it.