About 20 years ago, i returned to India before starting on a job I had accepted while I was graduating. I had completed the paperwork for processing a H1 (work visa) but wanted to pick it up in India (at the cost of a risk of not being granted one at the last minute) because a change of status to H1 from F1 while still in the US would not allow travel out of the country until a permanent resident status was achieved . In any case, here is the relevant part here:
When I had the H1 visa, the US Consul refused to issue the H visa for my daughter (born in the US but was added on to our Indian Passport) , saying that she must travel on a US passport and issued her one. I never saw the logic, but did not argue,, and went along.