What work or stories have been told of the great work of John the Beloved after he was translated? ALso if he tarries here on earth, does he go back and forth to Heaven and earth?
There are stories told, but there aren’t as many as one might think for very good reason. The most well known story is when he appeared with Peter and James to restore the Melchizedek Priesthood.
One of the reasons that there have not been more “stories” of John the Beloved is because he was commissioned by the Savior to go to the lost tribes of Israel and prepare them for their return to the lands of their inheritance and to the Restored Church. Reference to such facts may be found in the article by Hoyt W. Brewster, quoted below—
“One of the basic beliefs of Latter-day Saints is the “restoration of the Ten Tribes” (Articles of Faith 1:10) temporally to their homeland in Palestine and other places and spiritually to the gospel of Jesus Christ (3 Ne. 21:27-28; Mill M, 324). In 1831 the Prophet Joseph Smith declared that the Apostle John was then with the lost tribes preparing them for their return (HC 1:176, footnote). The scriptures tell us they will return from the north countries (Jer. 3:18; 16:14-15; 23:7-8; 31:8-9; Ether 13:11; D&C 133:26-34)”.(Hoyt W. Brewster, Jr., Doctrine and Covenants Encyclopedia , p.589)
As for him going back and forth between Heaven and the Earth… That would not match up with what I understand the word tarrying to mean. I don’t think he does. Still if there was a need he seems like he would be a logical choice to do so.