So a little update. Two chat support agents at Emirates told me that my baggage allowance was 30kg on K6 and 40kg on EK, but nothing on any of my tickets, itineraries, etc. confirmed this. In the app and on all documents my baggage allowance was 30kg for all segments.

When I got to the check-in counter for K6 and they looked up our tickets we were told I had a 25kg allowance. They were only willing to pursue researching it further because my pdf ticket said 30kg. The entire check-in process took more than twenty minutes while the K6 staff tried to confirm that we had a 30kg allowance, but eventually they did. They were very nice but I would not be confident this would happen every time.

The real issue in this situation is that I had two EK flights of 7-8 hours each and one 1 hour K6 flight. Both EK flights were upgraded to business and the K6 flight did not offer a business class ticket, so the baggage rules of our original tickets, Economy Flex 30kg, applied to the entire journey, rather than the 40kg rules that would have applied if they used the "most significant carrier" model. This is obviously annoying because we lost one of the benefits of upgrading to business class.