Hoster dies in 299 AC and is succeeded by Edmure.

How did Tully Hoster die?

Hoster Tully was the former Lord of Riverrun, head of House Tully, and Lord Paramount of the Trident. He was the father of Catelyn Stark, Edmure Tully, and Lisa Arryn. He had one brother, Brynden Tully, better known as the "Blackfish." ... His son, Edmure Tully, took his place as the Lord of Riverrun.

Who kills Lord Tully?

I stand before you as one of the senior lords in the country, a veteran of two wars. I'd like to think my experience has led to some small skill in statecraft and I understand that– Edmure Tully, nominating himself as Lord of the Seven Kingdoms, before being cut off by Sansa Stark.