The procedure is quite simple in principle: each new contributor confirms their permission to use their code under two licenses—MIT and GPL3. After a while, the entire codebase becomes dual-licensed. At that point, I leave it as is and (tentatively) copy the MIT portion to another location. Yes, the derived work will still remain under the dual license, and therefore under the GPL3 as well, but no one can stop me from copying the MIT portion, which, by coincidence, will cover the entire project. This isn't relicensing; it's rewriting it under a dual license.
It does seem to be quite some (very boring) work. I'm curious about your reasons for taking it up. Is it more of a principles kind of a thing or do you see it could make a difference to the project in the end?
At the very least, the ability to use the code as a library. A library under the GPL3 is unlikely to be popular.