Thank you for your answer. I also own a N64 NUS-FRA RGB one so the Elgato Cam Link is not an option.
I asked “My life in gaming” on facebook and they gave me those products as a viable solution.
StarTech.com USB 3.0 HDMI
and also the Epiphan brand but I did not find a device which also captures audio.
I do own a board with analogue audio output so Hdmi capture cards with no analogue audio is not a solution.
However, I found something interesting, the DVI + analogue audio to HDMI converters. Do those devices add lag ? Do I have to use a dvi + audio splitter to record ?