My apologies.
I didn't realize that rules were different for tickets originating in the US. Would I have been correct if the OP's trip was a) originating from outside of the US and b) on a single ticket, as in the examples on LH baggage rules
here?
When booking a journey with several airlines the free baggage regulations of the airline which flies the
most significant route of the journey (usually the longest route geographically) under whose flight number this route has been booked apply. For these purposes it is irrelevant which airline actually operates the flight.
The baggage regulations apply in each case up until the moment you break the journey for more than 24 hours (make a
stopover).
In any case, as
findark mentioned, the end result is the same, right? OP should get 3x 32kg/70lbs.