Today I will explain the fifth freedom flights for Ethiopian Airlines(ET), an Ethiopian airline that belongs to Star alliance. It is quite complicated route though, let me explain what it is like.　
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BLZ – HRE – LUN, LUN – BLZ, HRE – BLZ

HRE <-> BLZ(300 miles)
21SEP – 28SEP
C : JPY155,000 –
HRE -> LUN(250 miles)
21SEP – 28SEP
C : JPY115,000 –
LUN -> BLZ(435 miles)
18SEP – 25SEP
C : 97,400 –
ET34

It is all outside of Ethiopia, and the route changes depending on the date.　

ADD – LFW – GRU

LFW <-> GRU(3811 miles)
ET506/507
19OCT – 27OCT
C : JPY350,770 –

ADD – LFW – EWR

LFW <-> EWR(5181 miles)
ET508/509
17NOV – 25NOV
C : JPY430,850 –

ADD – HKG – ICN

HKG <-> ICN(1284 miles)
ET608/609
20OCT – 27OCT
C : JPY74,650 –

ADD – BKK – KUL

BKK <-> KUL(754 miles)
ET618/619
21OCT – 28OCT
C : JPY29,780 –

I think it is worth checking for because you might find quite a good price.

ADD – HKG – NRT

HKG <-> NRT(1842 miles)
ET672/673
21SEP – 28SEP
C : JPY100,770 –

ADD – ARN – OSL

ARN <-> OSL(240 miles)
ET714/715, ET1692, 1694/ET1693, 1695

It seems you cannot buy only this route.

ADD – VIE – ARN

VIE <-> ARN(800 miles)
ET724/725
21OCT – 28OCT
C : JPY86,240 –

ADD – KGL – BJM

KGL <-> BJM(109 miles)
ET806/807
19OCT – 26OCT
C : JPY52,780 –

LUN – HRE – ADD

LUN -> HRE(250 miles)
ET863
22OCT – 28OCT
C : JPY80,250 –

ADD – HRE – LUN, LUN – ADD

HRE -> LUN(250 miles)
ET873
24OCT – 30OCT
C : JPY : 117,490 –

ADD – LUN – HRE, HRE – ADD

LUN -> HRE(250 miles)
ET893
22SEP – 29SEP
C : JPY58,170 –

Only departures from ADD to HRE will call at LUN.

ADD – BKK – SIN

BKK <-> SIN(875 miles)
ET1306/1307

It seems you cannot buy the fifth freedom flight route only.

NRT – HKG

NRT <-> HKG(1842 miles)
ET1400/1401
21SEP – 28SEP
C : JPY100,770 –

Conclusion

There are lots of routes for LUN – HRE.

Also almost all of them are irregular flights.

Still I think you can keep checking the BKK – KUL route.