I think the obvious problem here is that secular governments do not exist to further the goals of the Church, and most Christians agree. John Paul II fought against communism because, while equitable distribution of goods free from the pursuit of money is a noble Christian cause, it is fundamentally disrespectful to the dignity of the human being, as it violates the freedoms of those under communist rule.

While open borders may be a great thing, as it welcomes the oppressed and suffering and is absolutely in line with Christian teaching, doing so can cause a burden on the citizens of whatever nation chooses to adopt that policy. It is not in line with Christian social teaching to obligate people to follow biblical teaching.

Philemon 1:8-9 NABRE

Therefore, although I have the full right in Christ to order you to do what is proper, I rather urge you out of love

Unless the citizens of a nation welcome such a policy, open borders, to the degree that they are injurious to the citizens, is very much an open question in Christian thought.