Ottoman authority over Muslims in other regions was based on the idea of pan-Islamism, which (to the best of my knowledge) emerged in its modern form in the late 19^th century. It positioned the Ottoman ruler as the caliph (khalifa), claiming both civil and religious authority over the global Muslim community (ummah).

In this period, the impact of pan-Islamism was minimal in British India, though not much historical work has been done for this period. It was limited to showings of sympathy towards the Ottomans in conflicts such as the Greco-Turkish War of 1897 (which was still enough to alarm the British). This sentiment continued to grow, with the presence of Jamal al-Din al-Afghani, a leading pan-Islamic thinker, from 1879 to 1882 acting as a catalyst. As an example, public preachings (kuthba) in the princely state of Hyderabad had been read in the name of its Muslim ruler, the Nizam, since 1857. By 1914, it was instead being read in the name of the khalifa (though the exact date of the change is unknown). This shift would prompt Lord Hardinge, the British Viceroy, to appeal to the Nizam in 1914:

I accordingly call upon Your Highness to stand forth as the spokesman and leader of the Mahomedans in India and to declare that England's cause is both right and just.

But it was only with Ottoman defeat in the First World War, and the Treaty of Sevres, that this sentiment was mobilized politically, creating the Khilafat Movement, which then blended with the broader Non-Cooperation Movement. It was based on the idea that the authority and existence of the khalifa was being threatened, and primarily demanded a recognition of the authority of the Ottoman ruler.

Sumit Sarkar's division of the Khilafat movement into a moderate (largely urban, led by elite Muslims, represented by the Central Khilafat Committee) and a radical (largely rural, led by middle-class Muslims and religious scholars ulama) strand can also help us get at pan-Islamism itself. As with the example of Hyderabad, pan-Islamism (and thus, a positive stance towards the Ottomans) was a recent phenomenon in the subcontinent. Its rise was concomitant with the rise of an educated Indian Muslim class disconnected from traditional Islamic education in a madrasa; and an Urdu vernacular press which helped circulate pan-Islamist ideas. While the former helped the movement gain national structure, it was the latter that shaped it. A prime example would be the Moplah revolt in 1921, which saw the idea of 'khilafat' being turned into a general anti-landlord (who were mostly Hindu, and some British) sentiment by the Moplah Muslims. This suggests that pan-Islamism in India was not a purely religious phenomenon, but had significant political and cultural elements.

Of course with Ataturk ending the caliphate in 1924, the entire thing tapered off.

Sources: Modern India by Sumit Sarkar (1989, Palgrave Macmillan) as a general introduction to the period and the movement; Reaping the Whirlwind by Margrit Pernau-Rei (September 1999, Economic and Political Weekly volume 34) for developments in Hyderabad.