Objection 1. It would seem that the indulgences granted by the Church profit even the dead. First, on account of the custom of the Church, who orders the preaching of a crusade in order that some one may gain an indulgence for himself and for two or three and sometimes even ten souls, both of the living and of the dead. But this would amount to a deception unless they profited the dead. Therefore indulgences profit the dead.
Objection 2. Further, the merit of the whole Church is more efficacious than that of one person. Now personal merit serves as a suffrage for the departed, for instance in the case of almsgiving. Much more therefore does the merit of the Church whereon indulgences are founded.
Objection 3. Further, the indulgences of the Church profit those who are members of the Church. Now those who are in purgatory are members of the Church, else the suffrages of the Church would not profit them. Therefore it would seem that indulgences profit the departed.
On the contrary, In order that indulgences may avail a person, there must be a fitting cause for granting the indulgence [Cf. 25, 2]. Now there can be no such cause on the part of the dead, since they can do nothing that is of profit to the Church, and it is for such a cause that indulgences are chiefly granted. Therefore, seemingly, indulgences profit not the dead.
Further, indulgences are regulated according to the decision of the party who grants them. If, therefore, indulgences could avail the dead, it would be in the power of the party granting them to release a deceased person entirely from punishment: which is apparently absurd.
I answer that, An indulgence may profit a person in two ways: in one way, principally; in another, secondarily. It profits principally the person who avails himself of an indulgence, who, namely, does that for which the indulgence is granted, for instance one who visits the shrine of some saint. Hence since the dead can do none of those things for which indulgences are granted, indulgences cannot avail them directly. However, they profit secondarily and indirectly the person for whom one does that which is the cause of the indulgence. This is sometimes feasible and sometimes not, according to the different forms of indulgence. For if the form of indulgence be such as this: "Whosoever does this or that shall gain so much indulgence," he who does this cannot transfer the fruit of the indulgence to another, because it is not in his power to apply to a particular person the intention of the Church who dispenses the common suffrages whence indulgences derive their value, as stated abovek2(27, 3, ad 2). If, however, the indulgence be granted in this form: "Whosoever does this or that, he, his father, or any other person connected with him and detained in purgatory, will gain so much indulgence," an indulgence of this kind will avail not only a living but also a deceased person. For there is no reason why the Church is able to transfer the common merits, whereon indulgences are based, to the living and not to the dead. Nor does it follow that a prelate of the Church can release souls from purgatory just as he lists, since for indulgences to avail there must be a fitting cause for granting them, as stated above (Question 26, Article 3).
Source: Summa Theologica (New Advent)