Upvote:0
AIUI (this is based on what I have read online, I am not an expert)
First you need to figure out if your children are dual citizens. Many but not all children born abroad to a US parent will be US citizens.
If your children are US citizens then they have the right to live in the US as long as they like. They should get US passports to allow them to freely leave and re-enter the US.
Secondly understand there is a difference in the US between "Visa" and "Status", similarly there is a difference between and "ESTA" and a "Visa waiver program status". Once your children are in the US expiry of the ESTA is irrelevent and extending it won't help, what matters is the expiry of the status granted when they entered the US.
Normally aliens admitted under the Visa waiver program are not allowed to change or extend their status. It appears there is an exemption for close relatives of a US citizen but it is not clear to me whether that exemption only allows adjustment to permanent residence or whether it allows the alien to remain in non-immigrant status.
One thing you don't say is whether the other parent is cooperative and contactable. I know that getting a US passport for a child normally requires the consent of both parents, I'm not sure if the same applies to adjustments of status. If the other parent is uncooperative or uncontactable then it is almost certainly time to get a lawyer involved.
Upvote:8
ESTA is not a visa. It's a travel authorization for the US. If you arrive in the US by plane or boat, on a Visa Waiver, you need this to board the plane or boat. It doesn't say anything about the length of stay in the US.
Your kids are in the US on a Visa Waiver, which is valid for 90 days. The ESTA is valid for 2 years. If you want them to stay longer than the 90 days, you'll have to work out what visa they can apply for. But it has nothing to do with the ESTA.