Madhwas claim a unique thing which as far as I know, no other Hindu denomination claims explicitly that
- Lord Vayu was first born as Lord Hanuman to serve Lord Rama in Treta Yuga.
- Lord Vayu was then born as Bhima to serve Lord Krishna in Dwapara Yuga.
- Lord Vayu was then born as Sri Madhwacharya to serve Lord Veda Vyasa in Kali Yuga.
The first Avatar of Sri Vayu is Hanuman, son of Anjana Devi. His wonderful feats and service to Lord Rama are described in the RAMAYANA.
प्रथमो हनुमान् नामा द्वितीयो भीम एव च |
पूर्णप्रज्ञः तृतीयस्तु भगवत्कार्य साधकः ||
Prathamo Hanuman Nama Dwiteeyo Bheema Eva cha |
Poornapragna Treetiyastu Bhagavat Karya Sadhakaha ||
The second Avatar is Sri Bhimasena. In Dwapara Yuga, Sri Vayu Deva appeared as Bhimasena and played the most important part in the destruction of the enemies of God. In the entire Mahabharata, Bhimasena played the most important role. He is the one who never slipped from the path of righteousness and served Sri Krishna.
Thus, he served God as Hanuman in Ramayana, as Bhima in Krishnavatara and, as Madhwa, served Lord Vedavyasa Devaru. He is the greatest spiritual benefactor of mankind. This, in short, is the story of Avtara Traya - the three avatars of Lord Vayu.
What is the scriptural source for 1 and 2? And is there any scripture which predicts 3?
Scriptures here means authentic Hindu scriptures like Vedas, Agamas, Smritis, Puranas, Ramayana, Mahabharata and Bhagavad Geeta which are ancient.