"Can the souls in purgatory be used as a powerful intercession to influence a safer end to the situation of the illegal migrants marching to our southern border? And, why can they intercede on our behalf, but not on their own behalf?

It is not the tradition of the Church to pray to the souls in purgatory. Just pray to Our Lord and Our Lady or the saints to bring a safe end to the illegal immigrant caravan.

Saint Alphonsus in his work the “Great Means of Salvation”, chap. I, III, 2, after quoting many renowned theologians as favourable to his opinion, concludes: “so the souls in purgatory, being beloved by God and confirmed in grace, have absolutely no impediment to prevent them from praying for us. Still the Church does not invoke them or implore their intercession, because ordinarily they have no cognizance of our prayers. But we may piously believe that God makes our prayers known to them”.

From the book My Catholic Faith by bishop LaRavoire Morrow:

All the souls in purgatory will go to heaven some day; they will stay in purgatory only as long as they have not atoned for all their sins.

The poor souls cannot help themselves, for their time for meriting was ended at their death. They cannot therefore merit anything to satisfy for their sins.

This is why we who can still merit by our good works should give some of them as suffrage for the poor souls, so that they may soon be delivered from their prison. We have the special obligation of helping with our prayers and sacrifices the Souls of our dead relatives, friends, and benefactors.

Although they cannot merit anything for themselves, the poor souls intercede for us with their prayers to God.

Thus if we help them they repay us by their intercession. No one who has a devotion to the holy souls in purgatory has ever asked for their intercession in vain.