A lot of Catholics are familiar with St. Gertrude's prayer for the souls in purgatory given her by our Lord. According to tradition. the Lord promised her that 1000 souls would be released from purgatory each time it is said devoutly. The prayer was extended to include living sinners as well. Here are the words:
"Eternal Father, I offer Thee the Most Precious Blood of Thy Divine Son, Jesus, in union with Masses said throughout the world today, for all the holy souls in purgatory, for sinners everywhere, for sinners in the universal church, those in my own home and within my family. Amen."
I am thinking that the reason this prayer is so effective in releasing souls from purgatory is that these souls are already heaven-bound and therefore no free will exists to interfere with our prayer request. In the case of praying for the living, the Lord will have to bring their free will in line for our prayer to be answered.