The thing that caught my attention is, when a person with foreign citizenship commits a crime on US soil, we arrest them. But, depending on the crime, we will usually deport them into the custody of their own country, and that country will apply their laws and legal system to their citizen.
An American can be thrown into a foreign prison for breaking their laws while in that country. But if they come home before the foreign police can arrest them outside the US, those cops can't just show up at the house in the US and take them back for trial and punishment. That country must go through the extradition process, assuming they have an extradition treaty with the US. Even then, the US may decide to deny extradition. All of that, because you are subject to the jurisdiction of the US, not the country you were a visitor in.
That pretty much describes what's meant by "jurisdiction" in my mind. Which means I agree that illegal immigrants and any of their children born in the US are to be deported as foreign citizens, because illegal immigration is breaking the law. These immigrants/aliens never officially renounced their non-US citizenship and never applied to become a naturalized US citizen.