Do avatars remain young or become old? Are there any descriptions of Rama , Krishna or any other avatar when they become old(aging process) and unable to walk etc.. Or is it that they complete their lila remaining young throughout the life.
No. The incarnations of Vishnu lie Sri Krishna does not age. To reply to this question, Anirvan (Swami Nirvanananda Saraswati, very renowned saint and scholar who translated Sri Aurovindo's Life Divine to Bengali) writes:
Look at one phenomenon of nature. The flow of Prana flow both ways in tide and ebb. Upto youth its basically tide flow (rajas), then ebb from youth to old age (tamas). In the youth, praana can realize that the time of ebb is coming. So it wants to keep on flowing by procreation. Taking food and procreation --both are 'jivana-yoni-prayatna'.
The Purushottama Avataras like Sri Krishna is samasti-bhuta.He is Bhuta-Maheswara. In Him, the tide always balances the ebb (Sattwa-Guna). Srimad-Bhagavatam terms this as "Atmanyavaruddhasaurata" bhava of the "Purusha".
So Sri Krishna is "yuvaaa aakumarah" in the language of the Vedaas.(Quoted in Sri Sri Omkar-Sahasralipi, Kinkar Madhav Ramanuja jeo (edited),vol. 2, page 368).
Sri Krishna says in Gita:
7.29 Those who strive by resorting to Me for becoming free from old age and death, they know that Brahman, everything about the individual Self, and all about actions. [They know Brahman as being all the individual entities and all actions. This verse prescribes meditation on the alified Brahman for aspirants of the middle class. Verses beginning with the 14the speak about the reaization of the unalified Brahman by aspirants of the highest class.]
He who can give liberation from old age is of course never old Himself. Sri Chaitanyadeva refers to Him as "Chira-Kishora".
The dhyana-Mantra of Sri Krishna also portrays Him as ever -youth:
Sat-Pundarikanayanam meghaavam vaitytaambaram/Divyalankaararanopetam gopagopanagnaavitam--- His eyes are like beautiful lotus and He is encircled by the gopas and gopis.(Stavakusumanjali, Udbodhan,Swami Gambhirananda(ed),page 123-4).
Kavi Karnapura describes Sri Krishna as
tribhangalalitam jane jaganmohanam" implying the One having Beautiful Tribhanga -form Who charms the entire world (ibid page 265).
We must remember that incarnations of Vishnu do not have material bodies. Their bodies are Divine. As Sri Krishna mentionsnins in Gita:
4.9 He who thus knows truly the divine birth and actions of Mine does not get rirth after casting off the body. He attains Me, O Arjuna.
9.11 Not knowing My supreme nature as the Lord of all beings, foolish people disregard Me who have taken a human body.
Brihatbhagavatamrita also says that Sri Krishna was ever youth He was "Kaishoragandhih". This was quoted by Sri Sanatan Goswami who explains it by saying that Sri Krishna is always the beautiful youth thoroughout His life.Radhagovinda Nath has sited these in explaining the different aspects of an Avataara( Mahaprabhu Sri Gauranga, Radhagovinda Nath, Sadhana Prakashani, page 146-148).