usa citized giving birth in mexico

My 6 months pregant 27 year old daughter, USA citizen, living in Mexico wants to give birth in Cabo San Lucas. She does not have a visa nor work permit for Mexico. She's living with her Mexican boyfriend and his parents. She wants to have baby in Mexico and thinks the baby will have dual citizenship rights. She plans to live in Mexico for a few years and then live in the States when child is eligible for elementary school.

Questions: If baby is born in Mexico, what immediate USA rights does the child have? Will my daughter face legal problems traveling back and forth from Mexico to the USA?

Generally, what kinds of issues and problems does she face by not traveling by car to the USA, giving birth in the States and then traveling back to Mexico to live?

Thank you for all your responses.