Per a Kyodo wire report in Japanese, the new “zairyu card” proposal (detailed here and here) coincides with an MLIT proposal for an updated resident registration (juminhyo) system, which will include foreigners for the first time.
This is perhaps the best part of the entire change. For years now, foreigners have been booked in a separate alien registration system, which creates all sorts of hassles: their Japanese spouses officially appear to be single, and the foreigner can’t verify their identity using the same means as locals (which confuses inexperienced clerks all the time).
So now the picture is a bit clearer. When a foreigner moves, they will update their juminhyo at city hall just like a Japanese person would, and this update will be forwarded to Immigration for recording in the zairyu card system. It still isn’t clear how extensions and changes in immigration status will work (the same in reverse?), but this seems like a promising line of developments so far.