Can you try opening the browser console/network tab and seeing if you get any errors when this happens? That will help me dig into the logs and find out what's going wrong for you
I've just tested it successfully for me, and a little green toast notification in the bottom right should appear. Remaining on the same page afterward is normal
There's another flow for individual projects you can try by going to Transfer Chat here

Speculating: A chat can have a connected Vercel project and a connected Github repo, so it's possible that transfer is failing because one or more of your chats has connected resources that it can't either move or disconnect. If you only get the issue transferring all chats but can successfully transfer some chats, it may just be a single one that was causing the issue and if you can narrow down to that I can help debug and fix the issue