I’ve had that as well. That’s why you shouldn’t cancel on sight. I allow them to enter to see what they have in mind.
I carry car seats (portable are smaller and foldable used for travelling called mobi go… u can write it off on taxes
I used to bring a carseat. Between height, weight, age etc and 4 types of actual carseats, I eventually realized the liability was too great. If the carseat fails, we as the carseat owners are liable 
Yup, if the child gets hurt while in your car seat, Uber/Lyft’s insurance will not cover you in a personal lawsuit.
This is a simple thing ,…
First, doors are ALWAYS locked and I roll window down part way and ask for name on the app. DO NOT say their name. Then ask if they have appropriate car seat or booster for that child.
Parent has child with appropriate car seat/booster. Parent installs seat and properly secures child , gets in and we go. Make sure you have the parents install the seat and you have that on your dashcam.
Parent has child without car seat/booster … doors stay locked and I drive away and report it to Uber/Lyft.
I don’t drive too far away, so I can wait my 5 minutes to get cancel fee, unless passenger goes ahead and cancels. If they cancel, drive away.