Objection 1. It would seem that suffrages do not profit even those who are in purgatory. For purgatory is a part of hell. Now "there is no redemption in hell" [Office of the Dead, Resp. vii], and it is written (Psalm 6:6), "Who shall confess to Thee in hell?" Therefore suffrages do not profit those who are in purgatory.
Objection 2. Further, the punishment of purgatory is finite. Therefore if some of the punishment is abated by suffrages, it would be possible to have such a great number of suffrages, that the punishment would be entirely remitted, and consequently the sin entirely unpunished: and this would seem incompatible with Divine justice.
Objection 3. Further, souls are in purgatory in order that they may be purified there, and being pure may come to the kingdom. Now nothing can be purified, unless something be done to it. Therefore suffrages offered by the living do not diminish the punishment of purgatory.
Objection 4. Further, if suffrages availed those who are in purgatory, those especially would seem to avail them which are offered at their behest. Yet these do not always avail: for instance, if a person before dying were to provide for so many suffrages to be offered for him that if they were offered they would suffice for the remission of his entire punishment. Now supposing these suffrages to be delayed until he is released from punishment, they will profit him nothing. For it cannot be said that they profit him before they are discharged; and after they are fulfilled, he no longer needs them, since he is already released. Therefore suffrages do not avail those who are in purgatory.
On the contrary, As quoted in the text (Sent. iv, D, 45), Augustine says (Enchiridion cx): "Suffrages profit those who are not very good or not very bad." Now such are those who are detained in purgatory. Therefore, etc.
Further, Dionysius says (Eccl. Hier. vii) that the "godlike priest in praying for the departed prays for those who lived a holy life, and yet contracted certain stains through human frailty." Now such persons are detained in purgatory. Therefore, etc.
I answer that, The punishment of purgatory is intended to supplement the satisfaction which was not fully completed in the body. Consequently, since, as stated above (1,2; 13, 2), the works of one person can avail for another's satisfaction, whether the latter be living or dead, the suffrages of the living, without any doubt, profit those who are in purgatory.
Reply to Objection 1. The words quoted refer to those who are in the hell of the damned, where there is no redemption for those who are finally consigned to that punishment. We may also reply with Damascene (Serm.: De his qui in fide dormierunt) that such statements are to be explained with reference to the lower causes, that is according to the demands of the merits of those who are consigned to those punishments. But according to the Divine mercy which transcends human merits, it happens otherwise through the prayers of the righteous, than is implied by the expressions quoted in the aforesaid authorities. Now "God changes His sentence but not his counsel," as Gregory says (Moral. xx): wherefore the Damascene (Serm.: De his qui in fide dormierunt) quotes as instances of this the Ninevites, Achab and Ezechias, in whom it is apparent that the sentence pronounced against them by God was commuted by the Divine mercy [Cf. I, 19, 7, ad 2].
Reply to Objection 2. It is not unreasonable that the punishment of those who are in purgatory be entirely done away by the multiplicity of suffrages. But it does not follow that the sins remain unpunished, because the punishment of one undertaken in lieu of another is credited to that other.
Reply to Objection 3. The purifying of the soul by the punishment of purgatory is nothing else than the expiation of the guilt that hinders it from obtaining glory. And since, as stated above (Question 13, Article 2), the guilt of one person can be expiated by the punishment which another undergoes in his stead, it is not unreasonable that one person be purified by another satisfying for him.
Reply to Objection 4. Suffrages avail on two counts, namely the action of the agent ["Ex opere operante" and "ex opere operato"] and the action done. By action done I mean not only the sacrament of the Church, but the effect incidental to that action--thus from the giving of alms there follow the relief of the poor and their prayer to God for the deceased. In like manner the action of the agent may be considered in relation either to the principal agent or to the executor. I say, then, that the dying person, as soon as he provides for certain suffrages to be offered for him, receives the full meed of those suffrages, even before they are discharged, as regards the efficacy of the suffrages that results from the action as proceeding from the principal agent. But as regards the efficacy of the suffrages arising from the action done or from the action as proceeding from the executor, he does not receive the fruit before the suffrages are discharged. And if, before this, he happens to be released from his punishment, he will in this respect be deprived of the fruit of the suffrages, and this will fall back upon those by whose fault he was then defrauded. For it is not unreasonable that a person be defrauded in temporal matters by another's fault--and the punishment of purgatory is temporal--although as regards the eternal retribution none can be defrauded save by his own fault.
Source: Summa Theologica (New Advent)