
Is there any benefit for the kids to have US or European citizenship?
Many people move internally to US or Europe with new role. If their children are born there, the children will get the citizenship there. Does it help in long run? If yes, how? What are the downsides?

US passport is among the top 10 strongest passports in the world. An Indian passport is somewhere in the low 90s. That alone should be reason enough

Having kids in Europe doesn’t get them the citizenship of that country. There are two kinds of citizenships - Jus Soli (citizenship by right of birth) and Jus Sanguinis (citizenship by right of blood). For instance, India follows Jus Sanguinis citizenship, so a person born in India becomes Indian citizen only if at least one parent is Indian.
US follows Jus Soli, so if your child is born there they get US citizenship.
All European countries follow Jus Sanguinis so your child doesn’t automatically become their citizen just because s/he was born there.

Interesting, never knew that Birth Right Citizenship isn't applicable in the EU

If parents move back to India before the kid is no more a toddler, then I think having a native passport for the baby is really helpful in pursuing education, getting access to better govt services and loads of other strong passport values. Downsides, probably if the kid grows up in foreign land, it’s being subjected to loads of uncalled cultural shocks, often said ABCD kids in America.