I know this post is about the USA but in the UK this issue is high in the publics mind.
If you were seeking asylum why travel through five perfectly good countries before getting on a plane, train or boat to the Island of Britain?
The rest of the EU want us to take our" fair" share of asylum seekers but if you were running from tyranny wouldn’t you stop in the first safe country? Or is running to the one with the best free health care and hand outs more appealing?
I can get Mexicans as its right on the USA border so your always going to get people trying (lets face it every cowboy film I see where they are in trouble they run the other way ; ) ) but others that travel over oceans and continents away without legal authority?
They are just taking the Mickey. If they have nothing to offer your country in the way of needed skills and it is not the first safe haven from say persecution then I am a firm “sorry but no”.
I personally welcome people of all faiths, cultures and colours.
Britain is better as a country for its wide multi cultural society but we are a small island with finite resources .The influx of people from the widened EU where legal immigrants are perfectly legally working here has pulled on our finite resources ,what we don’t need is the further pull on the nations resources with illegal immigrants.

Its difficult to state this without coming across as some sort of closet racist or xenophobe and the guilt of saying no to people who are worse off than ourselves does leave me with a bitter taste in my mouth but I cant see any other logical way apart from saying no.