In 1943, the latter claimed to be the reincarnation of the Shirdi saint.
That sensational claim facilitated the rise to fame of the young Sathyanarayana Raju, alias Sathya Sai Baba (officially born in 1926).
That term broadly signifies the Deccan Plateau, a vast territory covering Andhra, Karnataka, and Maharashtra.
The enforced move south in 1329 included many Sufis of the emerging Chishti Order.
The original entity lived in Maharashtra, a Marathi-speaking state.
Whereas the proclaimed reincarnation emerged in Telegu-speaking Andhra, located further south.
In practice, three of the devotional movements involved do not favour this usage, not regarding Sathya Sai Baba as a priority.
The designation of "Sai Baba movement" represents an academic theory.