LAX
4 AA, QF
TBIT BA, CX, JL, QF, MX

JFK
8 AA, AY, MX
7 CX, BA, QF, IB
1 JL
4 RJ

ORD
3 AA, IB
5 BA, JL, MX, RJ

LHR
3 AA, BA, CX, AY, JL, QF, RJ
5 BA

I'm not sure how JL could move out of T5 at ORD as all international arrivals are processed there. It would make a lot of sense to move to T7 or 8 at JFK.
The only sane option there would be to move to at LHR would be T5. With BA not having enough space there for their own operations (with all the bussing to remote stands), why would JL even want to move out of T3?
A move into T4 at LAX? Why bother? TBIT is a 30 second walk away from T4. It's not like they are at the other end of the airport...