I have to admit that I don't know how interlining works across carriers: I have had experiences where I was fully checked in for the second itinerary when I checked my luggage, there have been other times where (like now) it seemed to be done manually.
So, the luggage is correctly tagged - however,
I'm wondering whether in cases like these there is any transfer of information onto the second PNR - ie are tags alone enough to get the luggage home, or will the luggage be "rejected" in transit, because although there is flight information, it does not appear on the carrier's system ?