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My kids are dual Danish and US citizens. My husband and I are permanent US residents, our kids being born here.
When we travel back and forth between Denmark and the US, we have been solely using the kids US passports.
Obviously, for entry into the US (and check in at the airport) they need to use their US passport, as they have no visa.
On arriving in Denmark, they have a automatic 90 day travel visa, so just showing the US passport have been enough.
There are a couple of other questions around this I explored on the site: Travelling with two different passports, Traveling with two passports which one to use and more, and all of them state that you have to use the passport of the country you enter if you hold that passport, and that in some cases you are required to.
However, we have not been doing this for kids, as I heard anecdotally that it can cause trouble when entering the US. A friend of mine with dual US/other citizen ship used his other passport when entering that country, and at some point he was asked on US re-entry where he had been, and why he had no stamps in his passport. He was told by the US immigration officer that you have to use the same passport for your entire travel.
Is this correct? Or should I use my kids Danish passports for entering Denmark?