I'd tend agree with your comment in the short term. Moving my daughter's child safety car seat would be a pain, for sure. Longer-term I would expect to see modular standards emerge for child safety car seats. You lock your child's seat into the standardized base that's hiding under a seat cushion in the back of the car. There are already stroller / pram systems that incorporate the child safety car seat, so I could expect that market to boom, too.
Moving things besides yourself is something a lot of people in urban areas already do with public transit. With a private autocar service it's more likely that you'd get more space than on a subway or bus, so I'd think it would be even easier to move more than just yourself. I would expect a rental option to let you keep the car with you, as opposed to it returning to the fleet when you reach your destination.