I feel as if this should be uncontroversial. Not only uncontroversial, but a universal practice, regardless of communal affiliation. Maybe it is and I'm just too inexperienced with wider Christian practice.
I want, as I seek to enter this, to put aside all predestinarian polemic. In the end, Barth could be right and God could have elected everyone -- we just cannot see it with our necessarily limited historical scope. I won't argue one way or another. What we must do, I think, is to pray for the salvation of all as if we can influence God in His saving work. I don't say that idly, knowing that God does all His good pleasure; however, I also see that we are called to pray for all, for the will of God is the salvation of all (1 Tim. 2:1-6) and that we are to save our brother and sisters, acting as an atoning sacrifice, by turning them from the error of their wanderings (Jam. 5:20). (That these passages can be interpreted only as a participation in Christ's Cross should go without saying.)
There is another reason why we should pray for the salvation of all: our own sin. Follow the Bible's narrative: Adam, though he has the possibility to not sin, does. As one of my Catholic friends put it recently, we shouldn't blame original sin for our own sin: Adam didn't need it as an excuse, neither do we. Through this sin, though, we introduce death as a necessary component of human existence. It becomes the fact of our existence: memento mori. No one can escape from it, for the link of communion which Adam shared with God had been severed. Since then, we are all born into death: not only born to die, but born in a state of corruption, violence, and misery. We then recapitulate Adam's sin, except that where he had enjoyed the vision of God, we enter the world in darkness and continue blind. We are creatures who were made to seek the Good, but in the absence of experiencing Him (or even knowing that He exists), we turn to all sorts of lesser goods and so turn every action into idolatry of some sort. Instead of emptying ourselves out for others, knowing that the life of God is our inheritance, we hold back out of fear of loss or, worse, take with force from others to secure our right to the good against theirs. Sin is seeking after a lesser good with fear, with ingratitude, and with violence. No human is exempt from this situation.
In light of this, how can we not feel pity for our fellow man? To vaunt ourselves up, as if we aren't capable of the same evil as they, is to forget our common slavery. To exalt ourselves is to forget our complicity in their sin: for many of the things taken for granted in our world are built off of the sins of others, and on top of those sinned against. How else shall we understand the Lord Christ's words "Judge not, lest you be judged"? Or "if you do not forgive your brother his sins, neither will the Father forgive your sins"? "Love your enemies"? And, at the heart of the paradox of the Christian Faith, "be perfect as your Father in Heaven is perfect"? Our God is, and always is, a humble God. If we are saved, we are filled with the love of God poured out by His Holy Spirit (Rom. 5:5), which means compassion for those who have not found this purpose of man's life. It doesn't not mean breathing threats of God's damnation on a recalcitrant massa damnata, but a humble plea to others to share in the liberation effected through the Cross. It also means ceaseless intercession before the Liberator, who has judged sin and death, to save all those under the cruel tyranny of the demons.
Lord, have mercy, and save us all.