This is the responsibility of the Department of Homeland Security (DHS) and its often carried by the ICE or Immigration and Customs Enforcement Division. There might be a phone call, workplace raid, check on the immigration status in jail, or perhaps a failed application for a green card, asylum, and US citizenship when a criminal offense or other reasons for removal come to light during the process.
There are consequences to not showing up for deportation when ordered to do so which can include late-night arrests or bars when trying to come back to the United States.
This will all depend on a number of factors including whether there was an order in the past, how long the person has lived in the U.S., whether there is persecution in the deportees country of origin, etc.
The Immigration and Customs Enforcement (ICE) should make sure that the individuals parental rights are protected. ICE authorities should be aware of this situation and perhaps the detainee can remain in a detention facility closer to the family court where he must appear either by video or in person.
This is not certain. A person caught after reentering the United States might be placed into removal proceedings again where you will have the option to apply for several forms of immigration or ways to stay in the United States lawfully. For instance, perhaps the individual can apply for cancellation of removal or for asylum.
When you dont appear for a hearing, a removal order will be issued by the Immigration Judge given that the government can prove you have been sent a written notice of the hearing. In other words, as long as the government proves they mailed you the notice, your removal order will still stand even when you fail to show up.
Even when the judge orders to deport you, you may still be able to appeal. Your deportation can be put on hold as long as you are pursuing your appeal within 30 days of the order unless you decide to wave your rights in court.
An attorney can investigate whether the charge is correct or not or if there are any circumstances that would warrant defending your deportation. Immigration laws can be quite complex and the attorney will help you figure it all out. For example, he or she can help you find out when the charge against you is false; argue if there are any qualifications for asylum, and help you presenting a cancellation of removal application.