Instead of making the long journey to the south US border, why don't all Latin American migrants go to the Falkland Islands because it belongs to UK

Perhaps because of the difficulty of crossing the South Atlantic? Plus The Falkland Islands are not part of the UK, they are a self governing British Overseas Territory so arriving there would at best get you considered for asylum in the Falklands. I suspect that there are not many people who would regard that as worth the effort and expense.

See https://en.wikipedia.org/wiki/British_Overseas_Territories