Explain how each of these empires came to control muslim lands, and describe each empire's influence in the area: mongols, ottomans, safavids, and mughals.

Mongols: During the 13th century that witnessed the Mongol expansion through most Eurasia, the branch that attacked the southwestern sector of the empire (middle east and central asia) was the Ilkhanate. The mongols managed to conquered most of current Iran, Iraq, Armenia, Azerbaijan, Georgia, Turkmenistan, Turkey, Afghanistan and Pakistan, and although their reing was short, it left a lot of descendant who would fight for control of the muslim lands in future centuries.

Ottomans: after the crusades, the Seljuk people managed to consolidate themselves in the anatolian peninsula and all the middle east, even defeating the Bizantine empire in the 15th century and expanding to north africa and east europe. They were the longest of the empires here described, and represented a muslim world power for hundreds of years.

Safavids: This was one of the most important Iranian dynasties, and ruled from the 16th to the 18th century. The were one of the main causes of Shia Islam (one of the religion's branches) expansion in the area, and enact a military excelence unparalleled by other muslim empires except the ottomans.

Mughals: Ruling india from the 16th to the 19th century, it represented the last indian gold age. It's first ruler was a descendant of the Ilkhanate leaders, and his empire expanded islam on India even further, under a flag of prosperity and economic growth like no other of it's time.