A question that has no definitive answer really. Law and morality are not always corollaries. Legally, they go back. Morally, they might stay, depending on perhaps a few variables, primarily a life-threatening situation from external forces if they return.
North Korean refugees arriving in the South is a good example of this kind of situation; the South lets them stay, knowing that upon return they will be killed or put into a labor camp in which they will most certainly die (also because they are essentially kin of the same blood and lineage, a strong moral obligation), where China just ships them back. China favors legality above morality and has a differing view on the value of individuals as humans with rights. The US has no such moral obligation to Mexico or Cuba, thus making it a legal question in their eyes, and that makes the answer simple. If it were Canadians coming down or Brits pouring over, I bet there would be a different attitude.
Hunger and poverty are not sound reasons to just go to another country without due process. There are systems in place for these scenarios, but they are typically reserved for a mass exodus, not individuals.
The "we all should go because the natives were here first" argument is a separate discussion altogether with its own set of unique and now bygone circumstances and has no part in this discussion.
Is this one of those threads where the question posed is heavy and dense requiring thoughtful answers or is it another one of those times where one will be ridiculed for thinking too much or posting a long answer? Seems bait-and-switchy to me, but it is a good topic.