Of course they will. Illegal immigrants, and the so-called “anchor babies” WILL self-deport. They are honorable, fair, and law-abiding and do not want to be branded as people that “jumped the line”.
They do not want the scarlet letter of law breaker and line jumper especially within their own communities. The few that don’t will be purged by their own social groups or remain outcasts forever in the shadows and truly without ANY country.
They WILL return to their country of origin and work just as hard there as they did in the US to make their home country “great again”. Many private interest groups in the US will assist them from a far. They will make their countries magnets for the disadvantaged and disenfranchised just like the US is today.
They will not need to be tracked down, arrested, embarrassed, disrespected, or cause anymore undue problems or expense. They WILL do the right thing. They would not want to live in a country or society where their neighbors would follow any other course.