It depends. Growing up on the farm, we never fed our chickens except when the ground was covered with snow. They totally free ranged and foraged for themselves. Our hens also only raised chicks in the spring and summer when there was plenty of food available.
If you are feeding your hens, the chicks will eat the same food, whether that is starter, grower, or layer. If your hens totally forage for themselves, the chicks will also forage for themselves. I've also found that chicks raised by a broody tend to forage more on their own, even if the hen herself eats a lot of feed. The chicks still eat the feed, just not as much.
It will depend somewhat on the time of year, the quality and quantity of forage available, and the habits of the hen. I always have feed available for them, not starter but grower since that is what my older chickens eat when young ones are around. Some broodies see that their chicks eat a lot of the grower. Some of my broodies do not depend on that feed much.