Here are various plots to IY/YYH crossovers that I don't have time to write. Please feel free to use these outlines to write your own story. Make sure to message me so I can read what you come up with.
In which Kagome becomes Genkai, and Inuyasha becomes Toguro, this is their backstory.
As the group tracked the jewel shards, they became infamous among demons. Naraku set up the first dark tournament five hundred years ago disguised as a human lord interested in betting. His real motive was to get Inuyasha killed and find the strongest demon for him to absorb. He intends to do this indefinitely, absorbing a new strong demon every year and becoming immortal. The winning team is told that they will each get a wish along with Naraku's part of the jewel. Kagome and Sango wish to never be forced to participate in such a tournament again. Miroku wishes for a bride to bear his children. Inuyasha wishes to be fully demon. Naraku then tries to absorb him, but Inuyasha is able to snatch the shards within his body to complete the jewel before he fully joins. He uses the jewel to become fully demon, and with his new power is able to destroy Naraku. Unfortunately, he has also taken part of Naraku into himself, becomes evil and gains the ability to absorb others. Kagome returns to her own time and becomes a powerful psychic. She hears nothing about Inuyasha again until she is old and her student is invited to the tournament. Inuyasha had become accepted by his brother once he became fully demon, and they along with Kagura, and two other demons formed a team to find worthy opponents. Some time along their journey they lose the jewel and it is used to grant the wishes of the winners of the dark tournament. I am not sure if Inuyasha betrayed his friends for power and demonhood, or if it was a genuine mistake that he became evil. Either way, both Kagome and Inuyasha die over the jewel and the circle goes round again. In order to get rid of Seshomaru like elder Toguro, Inuyasha could possibly absorb him. I think Kagura and Kurama would have a really badass fight.