They did in some cases. The Ottomans claimed the Caliphate after they defeated the Mamluks in the 16th century, but didn't put much effort into the claim for a long time. In the latter part of the 19th century they started claiming the title and promoting the idea.

There was a long standing tradition of Muslims doing their Friday prayers in the name of the Caliph. Indian Muslims started praying in the name of the Ottoman Sultans, which really freaked the British out. In the lead up to WWI the Germans and British both obsessed over this. People legitimately thought the Muslims of the colonial empires (in Africa, Asia, and India) would rise up in the name of the Ottoman Caliph. This belief was one reason the Germans acceded to an alliance with the Ottomans.

It was tradition for the Sultan to declare a Jihad whenever the Ottomans went to war. So a Jihad was declared for WWI and everyone waited with baited breath... only for nothing to happen. Muslims in the British, French, and Russian empires fought for those states and not the Ottomans.

In the end the Ottoman caliphate claims had a non-zero effect on efforts to create a unified Ottoman, Muslim national identity. Even though the Ottomans aggressively promulgated the idea of their Caliphate in the 19th century, nothing much came of it.

One interesting side note is the Star and Crescent. This was not an Islamic symbol, just an old symbol of the Ottoman house. When the plans for the nation of Pakistan were drawn up, the fact that the Muslims of India had spent the last century or so doing prayers in the name of the Ottoman Caliph led to a lot of sympathy for their symbols. So they adopted the Ottoman symbol on their flag. History is cool!