LH will put you on the next available flight option, and that could be:

FRA-ZRH-LAX on LX
FRA-MUC-LAX on LH
FRA-DEN-LAX on LH/UA
FRA-LHR-LAX on LH/NZ
FRA-IAD-LAX on UA
(or anything else with US/UA/AC that will get you there)

If possible, they will try to keep you on flights within the LH Group, so via MUC or ZRH is most likely (space permitting)....

Only if there is nothing within LH Group (LH, LX, OS, etc.) they will try within Star Alliance, and if there is no availibility there, they will put you on something outside Star Alliance (at least my experience).