Moreover, we have repeatedly addressed the Rulers of States, the heads of agencies, and all upright and cooperative men, urging upon them the need to consider and resolve the very serious problems of refugees and migrants, and, at the same time, to think of the heavy burdens which all peoples bear because of the war and the specific means that should be applied to alleviate the grave evils. We asked them also to consider how beneficial for humanity it would be if cooperative and joint efforts would relieve, promptly and effectively, the urgent needs of the sufferings, by harmonizing the requirements of justice with needs of charity. Relief alone can remedy, to a certain extent, many unjust social conditions. But we know that this is not sufficient. In the first place, there must be justice, which should prevail and be put into practice.
Likewise, from the first days of our Apostolic Office, we have directed our earnest attention to all our migrant sons, and we have been most anxious about their welfare, both temporal and eternal.
Pius XII
Due to the considerable proportions reached by the illegal migrant phenomenon, legislation in all the countries involved should be brought into harmony, also for a more equitable distribution of the burdens of a balanced solution. It is necessary to avoid recourse to the use of administrative regulations, meant to restrict the criterion of family membership which result in unjustifiably forcing into an illegal situation people whose right to live with their family cannot be denied by any law.
Adequate protection should be guaranteed to those who, although they have fled from their countries for reasons unforeseen by international conventions, could indeed be seriously risking their life were they obliged to return to their homeland.
St. Pope John Paul II
The Church has always said that nations have the right to control immigration but there are greater issues that can not be ignored...and the family is one of them. St. Pope John Paul II and many others have writings on balancing this. The Bishops are following that lead.
Just because the Church firmly stands against family separation does not mean She is for open borders. To conflate the two is a logical fallacy born out of a tribalism that puts party first. And that, left or right, is dangerous.