Solution

Just like the football, the air in the car also behaves inertly. The inertia of the air causes the fact that it moves in the driving direction (forward) while the car is braking. So the air is slightly denser in the front part than in the back part of the car. Thereby the pressure in the front is a little bit higher than in the back. Therefore the force, which is acting on the balloon from the front, is greater than the force acting on the balloon from the back. The forces are not cancelling each other out anymore (cf. picture below). The difference of these two forces is a resulting force which is directed to the rear end of the car while braking. This external force is counteracting the inertia of the balloon, so that it does not retain its previous movement but moves backwards (to the rear end of the car).