Pure BS. They are staying here. Most of them are not going home. In 1970 one in 21 in this country was foreign born; today it is one in 8, the highest it has been in 90 years; and within a decade it will be one in 7, the highest in our history.
Of the 40 million immigrants in the country in 2010, 13.9 million (±99,000) responded that they came to the United States in 2000 or later. That is, 13.9 million immigrants were in the country in 2010 who had come to the United States between 2000 and 2010. Of course, some of the immigrants who arrived during the decade would have died or returned home by 2010, so the actual level of new arrivals is somewhat higher. The reason that the immigrant population did not grow by 13.9 million is that roughly five million immigrants died or went home over this time period so the net change was slightly more than 8.8 million.