I have a car seat bag which the seat perfectly sits in the bottom of, and then fill it up with toys/swimming toys/nappies etc. Never had an issue.

This is the bag i have had for the last 4 years and probably been used 20+ times without issue. I even managed to squeeze 2 seats into it last year.

Only thing is that its quite bulky combined with all the other bags, pushchairs and kids themselves - a weight is lifted once its dropped at the oversize..!