If you compare the amount of movement the keys undergo in transport to the amount they move when you play the horn, it's a wonder they don't all fall off during a gig!
Seriously though, carting your horn around in the back of a car isn't going to do any harm...and in all the years I've been doing it the only thing that's ever fallen off is the lyre screw.
Horns are shipped with corks holding the keys down because pads can 'blow' ( expand ) in storage due to the absorbtion of moisture over a period of time, and the manufacturer has no idea how long the horns will be kept in storage before being sold. It's also a way of ensuring that poorly seated pads have more than half a chance of sealing when the punter opens the box.
Regards,