I currently fly one of the Winston Cup drivers. I have also done some contract flying for one of the big team operations. I much prefer flying for the driver. The vast majority of the Cup teams have team airplanes. Hendrick has a number of Beech 1900s, others have Brazilias, lots of King Airs. The driver's airplanes are their own individual responsibility, as a rule they are not paid for by the team.
This is by far the best opportunity I've ever had! We normally head to the track on Thursday. I usually leave the airplane and hitch a ride back on Ray Evernham's Brazilia, so I'm home for the weekend. Then, race morning (early) we load up on the Brazilia and head back to the track with the Winston Cup pit crew. For short distance races (like Richmond or Atlanta) I'll sometimes just fly back home.
We don't fly to the "local" races. Charlotte, Darlington, Rockingham, Bristol, and Martinsville. Each of those tracks has two races per year, so that means 10 weekends off during the season. Add to that Easter and Mothers Day, there's plenty of time away from the job.
These jobs are (and have always been) word of mouth and being in the right place at the right time. Over 90% of the airplanes are based either at JQF (Concord, NC), SVH (Statesville, NC), or HKY (Hickory, NC). You'd have to be in the neighborhood in order to know about these slots. Turnover is very low. Hope that helps.