I can tell you a little bit about the Ottoman-Mughal relationship, especially in the 16th century as these empires became increasingly aware of each other. I am not particularly knowledgeable about Safavid Iran and the Mughal Empire itself, but I know that their relationship was far from amicable. The Safavids continued the tradition of Iranian/Afghan empires to extend their reach into the North Indian river plain. The Safavids fought two wars with the Mughals in the 17th century as did their successors, the Afsharids, in 1738. The Persians won all of these wars but were never able to consolidate their gains. (Nadir Shah even sacked Delhi.) Maybe someone else can elaborate on the Iranian-Indian relationship of the time.
One might think that due to the fact that the Safavids and the Ottomans frequently fought each other, seeking an alliance with the Mughals would have been only logical for the Sublime Porte. However, when the Safavids began encroaching on North India in the 17th century, the Ottoman position in the Indian Ocean had deteriorated and they had become a minor player, peripheral to Indian affairs. That had not always been the case.
Just a century prior to the Safavid's first war with the Mughals, the Ottomans had quite a significant presence in the Indian Ocean, operating from their (recently acquired) bases in Suez and Yemen and were vying with the Portuguese for control of the Indian Ocean trade. Both powers were newcomers to the region: The Portuguese had circumnavigated Africa in 1498 and acquired Goa in 1510, the Ottomans had conquered the Mamluk Sultanate in 1517 and Basra in 1533 (IIRC), thus giving them direct access to the Red Sea and the Persian Gulf respectively. Both powers sent naval expeditions against the other's strongholds (for example the Ottoman siege of Diu in 1538) and raiding their trade routes, however generally the Portuguese kept the upper hand in warfare on the open sea due to their sturdier and larger ships. Trailing behind the Portuguese in actual military power in the Indian Ocean, the Ottoman relied on an extensive network of allies and client states stretching as far as the Aceh Sultanate, which they supplied with guns (and gunners) and military advisors for their fight against the Christian invaders.
Ottoman "soft power" around the Indian Ocean relied on supplying knowledge, firearms, and invoking their position as caliphs and protectors of the Holy Cities of Mecca and Medinah (hâdim al-haremeyn) and thus the hajj, the Islamic pilgrimage. States seeking the Empire's support acknowledged the caliphate by reading out the khutbah, the sermon at the friday prayer, in the name of the Ottoman caliph and recognizing his (mainly theoretical) suzerainty over all Muslims.
One local ally of the Sublime Porte in India in the 16th century was the Sultanate of Gujarat. At that time, the Mughals were still a rising power, having just conquered the North Indian plain in the 1520s, but a definite threat to Gujarat. Outside assistance against Mughal expansionism was thus very welcome in Gujarat. Gujarati elites frequently played Mughals, Ottomans, and Portuguese off against each other, turning to who ever seemed less intrusive or promising at the time. Sultan Bahadur of Gujarat (1526–1535 and 1536–1537) was the first to decidedly turn to Constantinople for support by claiming that his land was caught between "Mongols on land and infidels at sea" (referring to the Mughals, a Turco-Mongol dynasty, and the Christian Portuguese), and since his time there was an at times influential pro-Ottoman faction of Rumi origin at the Gujarati court.
While the relationship between the Sublime Porte and the Mughal emperor Humayun was already strained due to the the Ottoman-Gujarati connection, it grew worse under Humayun's son Akbar. (Yes, the famous one.) Under Akbar the Mughal Empire expanded into Central India and to the coasts. Gujarat was annexed in 1573, thus giving the Mughals their first large Indian Ocean port. Akbar declared himself protector of Indian Muslim pilgrims, appointing officials to help facilitate and finance the hajj for Indians, even sending out a lavish delegation of his household to Mecca with offerings to the inhabitants in his name . The Mughal emperor even began to style himself as "Padisha [great-king] of Islam" and "Caesar" – titles claimed by the Ottoman caliph.
As the Mughal Empire began to claim the title of preeminent Muslim power in the world or at least as an equal to the caliphate, Ottoman-Mughal relationship was increasingly antagonistic. However, with the deterioration of Ottoman power in the Indian Ocean, the Sublime Porte was less of a concern to Indian powers anyway. After the debacle of Mombasa in 1589 (where an Ottoman expeditionary force was lost entirely), the last of a series of defeats in the 1570s and 1580s, Ottoman presence in the Indian Ocean was diminished partly by own choice. Henceforth, the Empire's efforts were directed against the Safavids in the east and the Habsburgs in the west. The Mughal Empire rose to be the largest Indian empire since the Mauryans and the primary Sunni Muslim state in Asia.
tl;dr: Ottoman-Mughal relations were not very amicable most of the time. When the Ottomans were still prominent players in India and reaching out to the Muslim rulers of the subcontinent, the Mughals were still a rising power. The Ottomans allied with an enemy of the Mughals and lost their main ally to Mughal conquest later. Furthermore, Akbar the Great's claim to suzerainty over Indian Muslims and patronizing of the pilgrimage conflicted with the foundations of the Ottoman caliphate.
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/edit: Spelling, clarity, added a tl;dr.