The three things to avoid when transporting a cake are bumps/potholes, jiggling/vibrations, and sudden stops/starts.
Tier cakes must be transported in a FLAT space in your vehicle (cargo area, trunk, floor) with plenty of space for the box. Slanted seats do not work because they tilt the cake sideways, thus increasing instability.
The boxes for tier cakes are usually between 14" and 20" square - please make sure you have this much room cleared. Also, depending upon the design, the height could be too tall for the passenger front floor because of the console above. Please think about the size of your cake ahead of time and plan accordingly. You may need to borrow a friend's vehicle if you drive a small sports car with limited space.
Please clear personal items BEFORE arriving to pick up the cake, and assure no items are positioned where they could fall on/into the cake.
Cakes may not be left in the car to run errands without softening/melting when the temperature in the vehicle exceeds about 75°F inside the car.