My children have dual citizenship in a south american country. If wife was awarded children, does that mean they could possibly be allowed to go my BPDw home country?
My son also has dual citizenship. Even if your wife was not awarded the kids (I'm guessing you mean sole custody?), she could travel with them to her home country. If you think she's a flight risk, there are other things you can do.
Custody is:Physical custody
Legal custody
Then there is: Visitation
Then there is the stuff you and your ex figure out in a custody order, like travel to foreign countries with the kids. In my temporary order (which became permanent), I made sure it said in clear language that I could travel to my home country with S12 for two weeks. For me to do that, N/BPDx would have to sign a notarized letter of consent giving me permission to travel across the border. Even during my marriage, I was not able to travel with S12 unless I had that notarized letter of consent.
It might be different for your country. My country is Canada, and apparently there are enough parents trying to dodge custody issues that it's a fairly hard border to cross unless you have parental permission.
If you are worried that she might flee with the kids, or go there with your permission and not come home, check into the laws and see if the country is part of the Hague Convention. Maybe you can get a cheap consultation with an international lawyer who specializes in family law. If her country is part of the Hague, then it will have agreed to a set of protocols for what you can do if she flees. Most countries that are part of the Hague say that they will recognize the custody order where it was initially drawn up. The problem is that it's a very stressful and expensive process to get the kids returned, if you can find them.
Better idea is to set up an alert with US customs so that when she tries to cross with them, you'll be notified. Or, have it entered into your custody order that if she travels with the kids to her home country, she must post a bond that she gets back when she returns. If she doesn't return, then you have money to pay for an investigation into where she is. If she tries to enter her country and gets caught at the border, you could file a motion for contempt against her.
Do the kids have US passports? If not, you'll both have to sign the forms to get them for your kids. Not just now, but later too, when they have to renew their passports.
Sorry if that's confusing. Basically, you could get primary physical custody, joint legal custody, and the kids might see you 50% of the time, and they would still be able to travel to her home country with her -- if you want some say in whether she can do that, you have to include that item in your custody order.