AzhvArs are, in fact, not incarnations of any of the nityasUris as they are known to be right now. This has been a subject of great debate and discussion right from the times of svAmi nampiLLai. In fact, The Most Merciful periyavAchchAn piLLai starts off his commentary of AzhvAr's (note: wherever the word AzhvAr is mentioned alone, it refers to svAmi nammAzhvAr) tiruviruttam, by tackling a few opinions about AzhvAr logically, wherein he states that AzhvAr cannot be a mukta or a nityasUri who has been ordained by sarvESvaran to spread His word, because if that is the case, then AzhvAr being a personality who knew all the events of the past, present and future, would have waited patiently for His return to nitya-vibhUti. But we see that that is not the case, because right from tiruviruttam (1), ini yAm uRAmai (may we, (i.e.) both AzhvAr and the samsAris, not suffer this nature of being born repeatedly in this samsAram) upto tiruvAymozhi (10.10.11), avAvaRach chUzh (sarvESvaran who has surrounded me by quenching my desire of being united with Him), we can see that He has repeatedly requested sarvESvaran to eliminate the bondage of samsAram. Still, one may ask, "But AzhvAr himself states in many places that he has seen sarvESvaran. Since that is not possible for a nityasamsAri, how can he not be deemed as a nityAtmA?" But the reality is, though He has spoken of "seeing" sarvESvaran 10 times (in tiruvAymozhi (1.9.9), (2.8.10), (3.2.10), (4.7.7), (5.7.5), (5.8.1), (6.3.1), (9.4.8), (10.4.9) and (10.8.8)), all of these "sights" are those seen within the mind, as spoken by AzhvAr in periya tiruvantAti (28), nenjennum uTkaN (The inner-eye known as mind). Also, AzhvAr's longing to see sarvESvaran is echoed 20 times ((i.e) twice the number of times that he has "seen" sarvESvaran) in tiruviruttam (84), tiruvAymozhi (1.5.7), (2.4.2), (3.2.8), (3.8.4), (3.8.7), (3.8.8), (4.7.1), (4.7.9), (4.7.10), (5.8.4), (5.8.9), (5.9.5), (6.9.4), (6.9.6), (8.1.1), (8.5.1), (8.5.6), (8.5.10) and (9.4.1). Hence, due to his restlessness and haste in seeing sarvESvaran and his utter dislike when it comes to the nature of samsAram, we can conclude that AzhvAr is not a nityaSuri.
One may ask, "But these are all conclusions. Has AzhvAr said anywhere that He is a nityasamsAri?" Yes. In tiruvAymozhi (2.6.8) he says, mARi mARip palapiRappum piRantu (I have been born time after time). But does all of this undermine the glory of AzhvAr? No, because AzhvAr is clearly higher than all of the nityAtmAs, muktAtmAs and samsAris. This has been stated by svAmi nampiLLai in the mutal SrIya:pati of ITu-mahApravESam and also by azhagiya maNavALap perumAL nAyanAr in his AchArya hrdayam. Infact, nAyanAr goes on to elucidate the similarities between AzhvAr and sarvESvaran's eternal consorts and His consorts during His incarnations, sItA-pirATTi, His brothers viz. iLayaperumAL, bharatAzhvAn and SatrughnAzhvAn, great personalities like daSaratha, yaSOdA, prahlAdAzhvAn, vibhIshaNAzhvAn, tiruvaTi (mAruti) and arjuna, nityasUris like bhUmi-pirATTi and tiruvanantAzhvAn and great women like draupadI and kausalyA and also explains that AzhvAr stands out distinctly in his eternal love for sarvESvaran.
He is not of the ranks of nityasamsAris as well, being bestowed with faultless knowledge as stated by him in tiruvAymozhi (1.1.1), mayarvaRa matinalam aruLinan (He bestowed me faultless knowledge), thus being able to recite tiruvAymozhi as dictated within his heart by sarvESvaran. Due to all of these reasons, AzhvAr is to be deemed as a personality higher than both the nityasUris and the nityasamsAris.
Since the other AzhvArs are the avayavams (parts) of AzhvAr who is the avayavi (the one endowed with those parts), all of the above characteristics are true for them as well (with maybe ANDaL being an exception in citing periyAzhvar to be "garuDAzhvAn" (meimmaip peruvArthai viTTuchittar kETTiruppar (nAchchiyAr thirumozhi (11.10)).