Citizenship for the Sussex Children

I have always been curious how citizenship works for Prince Harry and Meghan's kids. Archie was born in the UK so I assume he has British citizenship, but Lilibet was born in the US and has US citizenship. Does she automatically get British citizenship due to her royal status. Would Archie have dual citizenship dual citizenship because Meghan has US citizenship. They seem to be able to travel easily so I wonder if they have some kind of pass or something.