I've been trying to figure out why they are coming here, seems from normal news sources its 50/50, some are escaping crime/corruption ridden areas and others are wanting a better life, with a better wages. Of course there is always going to be a certain percentage of any group that tries to criminally game the system, like most groups, this is probably a very low percentage.
While some of them are most certainly hoping to "live the American dream" of being able to work their way to prosperity, I think far too many are coming for the handouts. For those that want to escape crime and corruption, Mexico gave them an option, but it seems nobody wanted it. They could have settled in a land with similar cultural attributes and language, where despite all the negative news, there are large "middle class" areas not dissimilar to Anytown USA.
If nothing else, they could initially accept Mexico's offer and then apply to legally immigrate into the US if this is what they really want to shoot for. I still argue that 95% of this group does not qualify for asylum status, which to me generally means there is an immediate need to leave due to danger to life. In that case again, as per international agreement, you accept asylum in the first safe State you enter, which in this case is Mexico. When they said "no thanks" to the asylum offer, to me, at that point, they just became emigrants.