A former General Authority office was the Patriarch to the church. The responsibility of the one holding the office of Patriarch to the Church is to give patriarchal blessings. It was based on heredity starting with Joseph Smith's father Joseph Smith, Sr..
From Joseph Smith, Sr., the office passed on to Hyrum Smith and then, by birthright, William Smith, though he never lived worthy of the office.
It then went to John Smith, Father Smith's brother, and to Hyrum Smith's son, John Smith, who served faithfully for many years. Following his death his Grandson Hyrum Gibbs Smith served as the patriarch. Upon his death, for 10 years, acting patriarchs were called to the church out of the hereditary line.
Then Joseph Fielding Smith (not the same who was prophet) became patriarch. He was not in direct line. He asked to be released because of poor health though.
Then the office came to Eldred G. Smith who served for 32 years. It was a hereditary office. However there were many problems with the office. Must it go to Oldest worthy son or any member of smith family? Why a hereditary office. Eldred Smith rarely received conference assignments too. Would he have sealing powers too.
The patriarch to the church could give blessings to any member of the church whereas stake patriarchs could only give blessings to people in their stake (as noted above). He generally blessed people living away from a stake and toured the world doing it.
Now people without a stake patriarch can go to a neighboring one. However, Elder Smith, generally blessed others. Since he is still an ordained patriarch, he can and does give blessings. He just doesn't travel the world. It is unlikely, upon his death, that another patriarch will be appointed.
- “The Sustaining of Church Officers,” Ensign, Nov 1979, 18 - Eldred G. Smith made an Emeritus Patriarch