The doctrine acquired a financial apparatus. If the temporal consequence of forgiven sin can be remitted, and if the church holds authority to remit it from the merits of Christ and the saints, then an indulgence can be granted for prescribed acts of prayer, pilgrimage or charity, and applied to the dead as well as the living. By the late Middle Ages indulgences were being preached as a transaction, and the campaign in Germany that funded the rebuilding of St Peter's was preached in exactly those terms.
Luther's ninety-five theses of 1517 were written against this. Several of them assume purgatory rather than deny it; his initial complaint was that the practice made repentance a purchase and misrepresented what the pope could do. The polemic then radicalised, and the mature Protestant position rejected purgatory outright, on two grounds: that it lacks warrant in the canonical Scriptures, and more fundamentally that if Christ's atonement is sufficient and justification is a completed verdict, then no further satisfaction remains to be made. Trent responded in 1563 by affirming the doctrine while ordering the suppression of financial abuses and forbidding the sale of indulgences.
Eastern Orthodoxy occupies a third position that is easy to miss. Orthodox Christians pray for the dead universally and hold that the prayers of the church benefit them, but they reject the Latin framework of satisfaction for temporal punishment and the juridical accounting behind it, and the Orthodox delegates at Florence objected specifically to purgative fire. The Orthodox insistence is that the state of the dead is genuinely unknown and better left described in prayer than in doctrine.
The comparative point is that all three traditions do something for the dead. What separates them is whether that action is understood as affecting a defined process, and whether the church holds jurisdiction over it.