Hi may i share something i found while reading some articles about this issue.
I am also trying to find out the connectioms between the peverells, the slytherins and the gryffindors. After a long search i stumbled upon the family tree of the peverells. It's a long and extensive lineage intertwined with other known pure blood families (Blacks, McMillans, Prewetts and Malfoys to boot). I also saw a short family tree of the Slytherin family. The notions that Harry and Tom/Voldemort are distant relatibes is indeed clear since both are descendants of the Peverells. However, the line was cut off and was continued when the Gaunts emerged. Back tracking to the Tale of the Three Brothers, remember that Cadmus Peverell asled for the Resurrection Stone from Death to bring a loved one back to life. And since the "gift" he got didn't really did what he wanted, he committed suicide. At first I thought that ends the line in his side of the family, but then again, what if Camdus had a child to that woman before she died? That may have continued his line. Then a few generations after, Salazar Slytherin emerged and may have wedded one girl from the line. Based on the Slytherin family tree I got, Slytherin is not descended from the Peverells, rather he was married to one. Years passed then the Gaunts energed in the tree, and we all know what happened next.
As for Ignotus' line, it also has the same vague anscestry as Cadmus' since after him no one was mentioned for quite a long time, until the Potters appeared. Isearched and read a lot of HP Articles, even interviews of JKR herself and some other sites that she is connected to, but no such connections is seen between Godric Gryffindor and the Potters/Peverells. One things is certain, Ignotus had a son whom he passed on the Invisibility Cloak. The transfer from father to son maybe is seen there but there is a possibility that somewhere on the line the male side died, but a daughter was born. So, a mother to daughter passing of the Cloak was made until it is passed on again to a male descendant, and that is James Potter. I also tried to search for connections between the Evans and Gryffindor, but one article said that the Evans has a no significant role in the story.
This is just my understanding of how the relations between the characters are seen. No articles ever stated that Harry is indeed a descendant of Godric, we never knkw. We can only guess.