In his foreword, to James B. Gould's, Understanding Prayer for the Dead: Its Foundation in History and Logic (Cascade Books, 2016), Jerry Walls says:
The author distinguishes four kinds of prayers for the dead, and notes that the main Christian traditions have differed on the matter of which of these kinds of prayer are appropriate. The four kinds of prayer are for consummation, growth, purification, and salvation. While the first kind of prayer is most widely accepted and practiced, by many Protestants as well as Eastern Orthodox and Roman Catholics, the second and third types of prayer are accepted less commonly by Protestants, but are practiced by the Orthodox and Catholics. However, the fourth kind of prayer, for salvation, is generally rejected by all three traditions, on the ground that postmortem repentance and salvation are impossible.
As a Protestant who has written a book defending a doctrine of purgatory, including postmortem repentance, I am both intrigued by Gould's argument as well as attracted to it. Indeed, the early practice of prayer for the dead, particularly prayer for purification, was one of the factors that led to the eventual development of the doctrine of purgatory. The traditional doctrine of purgatory however, pertains only to persons who died in a state of grace, so postmortem salvation is excluded…As Jairus and his friends learned, death may not be the insurmountable barrier we think it is.
i) One question concerns the boundaries of Arminianism. Walls keeps moving the border stone. Does Arminianism have definable boundaries? What is out of bounds? How far can you redefine traditional Arminianism before it ceases to be Arminian? Jerry has a bunch of groupies who rubber-stamp whatever their guru says. Whenever he has forthcoming book or interview, they say "I can't wait!" They agree with him in advance of whatever he says. Nowadays, Arminianism seems to be harmonious with just about anything besides Calvinism.
ii) The appeal to Jairus is a bait-n-switch. His daughter wasn't even dead at the time Jairus dispatched his servants to solicit Christ's intervention. And even if she was, that would be a "prayer for the dead" is the sense of petitioning God to restore a decedent to life. That's completely different from a "prayer for the dead" in terms of purgatorial sanctification or postmortem salvation. Jerry's comparison is criminally equivocal.
iii) Whether prayer for the dead, in Jerry's sense, is permissible depends in part on your theology. It's not so much a question of directly challenging prayer for the dead, but challenging the underlying theology.
iv) In addition, there are disanalogies between intercessory prayer for the living and intercessory prayer for the dead. Much intercessory prayer presumes the liabilities of life in a fallen world. The kinds of harms and deprivations to which we're vulnerable in the here-and-now. Disease, poverty, suffering. Life in a fallen world is hazardous and precarious. Picking your way through a minefield.
But in classic Protestant theology, when Christians die, that takes them out of harm's way. They no longer have the same needs. They can no longer be hurt. They've put all that behind them. They leave the world of pain, danger, and suffering behind. That's very liberating. A huge relief. They are safe and secure in heaven. They no longer need intercessory prayer.
But for people like Jerry, the afterlife is an extension of the fallen world. Logically, if the lost can be saved in the afterlife, then the saved can be lost in the afterlife. If the psychological dynamic is fluid in one direction, why not the other?
v) If our prayers can facilitate postmortem salvation, why does the Bible never once command us to pray for the dead? Prayer is a huge part of Biblical piety. Both Old and New Testaments are chockfull of prayers and commands to prayer. If postmortem salvation is possible, if that actually happens, if prayer for the dead makes a necessary contribution to the salvation of decedents who wouldn't otherwise be saved, then there's nothing more important that you can pray for. So why the silence of Scripture?
vi) There's a major point of tension between belief in God's universal love and belief that death is the cutoff for salvation. But one can relieve a point of tension in either one of two different directions. Because Walls regards the universality of God's love as nonnegotiable, he makes whatever adjustments are necessary (e.g. postmortem purgatory, postmortem salvation, prayer for the dead) to relieve the tension.
Problem is, there's no evidence that his postulates are true. It's a third story conjecture resting on a second story conjecture resting on a first story conjecture. A skyscraper of wishful thinking.
vii) In addition, there's at least prima facie evidence that his position is contradicted by some passages of Scripture. And that's not confined to Calvinism. That includes Jansenism, Thomism, and Augustinianism.