R' Natan Slifkin examined the issue of transferring mitzva credit to other people, in the course of a broader discussion of benefiting people who have passed away, in a memorial lecture for his mother-in-law, entitled "What Can One Do For Someone Who Has Passed Away?" It is available for download in essay form here.
R' Slifkin points out (p. 3) that the idea of transfer of mitzva credit is dependent on the mystical concept of the product of mitzvot being supernatural spiritual emanations. This mystical concept is rejected by rationalist schools of thought in favor of a more practical concept of what mitzvot accomplish: "According to Rambam, mitzvos affect society, our intellects and our personalities;
other authorities express the function of mitzvos as creating a relationship with God.
Either way, mitzvos do not produce spiritual energy, such that one could transfer this
He cites (pp. 4 - 5) a collection of instances of Geonim, Rishonim, and Acharonim rejecting the idea of transferring mitzva credit and provides quotations of two Geonim, as cited by later authorities:
Rav Sherira Gaon, cited by the Rashba in his Responsa, Vol. 7 #539 (this and the next translation presumably by R' Slifkin, source links added for Mi Yodeya):
A person cannot merit someone else with reward; his elevation and greatness and
pleasure from the radiance of the Divine Presence is only in accordance with his deeds. Even if all the righteous people in the world were to seek mercy for him, and
all the righteous acts were to be done in his merit, it would be of no help to him…
However, most of the responsa is spent describing how another can help with their Mitzvos primarily by alleviating punishments through giving charity in the merit of the person or through a Tzadik praying for them. It then goes on to justify practices such as remembering the dead (Yizkor) based on this idea.
In this, like the Rashba, he rejects transferring reward. He does not address punishments, however he says that it does help to support and pay for someone to learn Torah, as the act of paying or even better supporting through removing distractions is the cause of the reward.