The expansion of the Ottoman Empire was a bit of an odd duck when it comes to accompanying mass migrations. They conquered much of South Eastern Europe, South Western Asia, and the old Arabic Caliphates in the Middle East and the North African territories. Some of the mass migrations, like those into Europe and Asia, are easy to see in the styles of buildings and mosques that exist to this day. Or in some cases don’t. Other mass migrations like those into and out of the Arabian and African regions are harder to track.
The exact reasons for this are relatively unclear. Did they move in and then leave again, or did they never move in more than a few basic administrators to watch over the conquered territories who were later absorbed into the general population? The predominant faith of both ethnic groups was Muslim by that time, so the religious wars that took place were more disagreements between the different flavors of Islam, and those flavors crossed ethnic borders. This all makes mass migrations in those areas a bit harder to track.
There are basic types of architecture that are famously Ottoman that can be seen throughout the old Ottoman Empire. Experts can easily recognize a surviving Ottoman government building, but the Turks themselves were far more willing to pull up stakes and move somewhere else than their European counterparts. And the modern Middle East and North African nations they once controlled exhibit a distinct lack of respect for or influence from their former rulers if you ask the modern day Turks.