I'll try to answer your question, sorry if it doesn't satisfy you. I'll focus on the late 18th to the early 19th century.

Well obviously you can't just leave the Empire willy-nilly, especially if you're a peasant reaya. If you want to travel or leave the empire, you have to obtain a travel document, either a mürur tezkiresi or bilet. The former is a simple travel document, required for all Ottoman subjects intending to travel outside of their hometown or village. Internal checkpoints were established and any individuals found on the open road without a mürur tezkiresi were liable for punishment. The mürur tezkiresi tended to be a small document, was generally printed on poor-quality paper, and contained only four lines of basic information on the traveler and his business (name, purpose of visit, intended route, and final destination). If you want to leave the empire, you can obtain (or counterfeit) mürur tezkiresi to leave your home and immediately head to Russia or to the Principalities of Wallachia and Moldavia, after which you can either stay or proceed to Russia, that is if you don't get caught or detained along the way

Another type of travel document is bilet, which has better chance if you're from the Balkans (especially Bulgarian) and wanted to leave the empire. Bilet was issued to migrants seeking refuge in Wallachian or Moldavian territory under certain circumstances. It was especially handy because it was issued according to Article 13 of the Treaty of Adrianople. In this article, the Ottoman and Russian empires agreed to allow free and unfettered migration of any and all Ottoman and Russian subjects wishing to resettle from the territory of the Russian Empire to the territory of the Ottoman Empire and vice versa for a period of 18 months. The first bilets were issued for Bulgarian migrants seeking to settle in Bessarabia and Moldavia. Just like mürur tezkiresi, bilet can be obtained from officials or by counterfeiting, which was quite rampant in the 1830s. Theoretically, Bulgarians who wished to change their settlement location within the Danubian Principalities, engage in a secondary migration to the Russian Empire, or return to the Ottoman Empire were obliged to petition relevant government authorities for transfer rights. In practice, however, once a Bulgarian migrant crossed the Danube into Wallachian or Moldavian territory, he or she enjoyed a freedom of movement that belied the ability of Russian, Wallachian, and Moldavian authorities to control migratory populations. So there you have it. Yay

These all however, relies on the possibility that you won't get caught or detained by Russian or Ottoman officials, which were there many especially by the Ottomans which adopted measures to stem emigration. For example, in May 1805, a large group of Bulgarian migrants attempting to cross the Danube into Wallachia were detained by an Ottoman customs official, stripped of their belongings, and ordered to return to their homes in Rumelia. In a similar vein, harbor officials in Danubian ports such as Tulça and Isakçı searched the holds of arriving ships for “illegal” Orthodox Christian migrants. Ottoman provincial governors in northern Rumelia dispatched raiding parties across the Danube River into Wallachia to seize and forcibly return Bulgarian migrant families. Passing through with counterfeit documents could also get you detained, for examples a certain Tahir Bey was detained and interrogated by Russian military authorities in Wallachia for attempting to travel through the Danubian Principalities on a fake document. You could also get blocked by Wallachian-Moldavian or Russian officials, though the Russians were way more lenient in their treatment of Balkan migrants, since migrants could be settled in lands and provide tax and production base.

Sources:

Selim III, Social Control and Policing in Istanbul at the End of the Eighteenth Century by Betül Başaran

Migration and Disease in the Black Sea Region: Ottoman-Russian Relations in the Late Eighteenth and Early Nineteenth Centuries by Andrew Roberts

I am not an expert on the subject but I can provide a relative example from the Jewish community involved in the textlile industry in Constantinople in the 16th century. The community was comprised by the greek speaking Romaniotes, the byzantine Jews who were conquered along with the rest of the byzantine Christians and the Sephardim, the iberian Jews who had willingly emigrated to the Ottoman Empire fleeing the Inquisition.

Because if their involvement in the textile industry which was considered strategic, they responded directly to the Sultan. So when an occasional epidemic of plague afflicted the city, they were not allowed to leave it because the Sultan could not afford having them not return. But this applied to the Romaniote and not to the Sephardim or at least not so strictly. Thus the Romaniote write to the Sultan lamenting this fact. Of course all these in the early centuries and certainly before the Tanzimat.

To continue with the subject of the Romaniote jewish community of Constantinople, a part of it was comprised by the Romaniote of Thessaloniki who were forcibly moved there - a widely used practice called surgun - to repopulate a sacked Constantinople after 1453. These certainly did not have the right to leave but still we do have internal movement like the Salonikan Sephardim participating on the revival of Tzfat in Israel in the 15th century.

So the answer is: depends on your trade and how valuable you were for the Empire, plus whether you had willfully submited to the Sultan or not. And of course this was more or less or not at all enforced through the centuries.

The aforementioned letters were presented by Eyal Tzinio, while information on the movements can be found in Nehama's History of the Jews of Salonikj.