The Catholic Catechism says immigrants have a "duty to obey the laws", as I cited at #9 (Link), and as you can read in context here (Link), Note that Paragraph #2240 stresses the duty to submit to legitimate authority and to pay taxes, and #2241 spells out the duty of immigrants to respect the country they are entering, obey its law, and assist in carrying "civic burdens."
That would be: the Church is against illegal immigration.
But some have not.
You might reasonably say, "OK, but what's the political policy of the Mexican Catholic Bishops? The US Catholic Bishops? The Catholic Bishops of the EU?"
Well, the US Catholic Bishops (USCCB) as a clerical bureaucracy has been notoriously more responsive to their liberal staffers' agendas than to the teachings of the Church; but then again, my view is that the bishops should go back to spending 100% of their time "bishoping," and the USCCB "as a clerical bureaucracy" should be abolished.
Can I get an Amen?