I'll try to answer your question, sorry if it doesn't satisfy you. To my knowledge, foreign Europeans fleeing to the Ottoman Empire to avoid dissent is quite rare, and the Hungarian flight to the Ottoman Empire is a unique case so I'll elaborate on that first

The Ottoman Empire is generally lenient towards refugees, and the Hungarians were no exception, even more as their outstanding feat of opposing the Austrian and Russian armies were well known in the Ottoman military circles and they brought numerous specialists that can be of use to the Porte, moreover the Ottomans wanted to increase their diplomatic reputation in Europe by sheltering romantic immigrants. However, taking some 3,700 Hungarian refugees could incur the wrath of both Russia and Austria, so the Ottomans had no choice but to repatriate most of them and wait for other Great Power's response for the remaining 1,500 held at the Ottoman border city of Vidin and Bucharest. France and Britain then voiced their support for the Ottomans, but the treaty made between Austria, Russia, and the Ottomans held the refugees back, as it's made clear that each country must hand over any "rebels" to their respective countries. However, there is an exception clause to this, where Muslim refugees can't be turned back by the Ottomans. So the Ottoman officials just plainly told them that. Most of them then converted to Islam, some genuine, some just a ruse and were allowed to enter the empire. Some of these men like your Jozsef Bem (as a side note, there is little evidence that he's appointed as a governor of Syria, however he still held a high rank in the Ottoman army) and Gyorgy Kmety entered the Ottoman army and fought in the Crimean war, and some just moved to other countries after things cool down

With that aside, now to your main question. I've addressed about Ottoman migration in general here, So i'll go in a bit of detail about Ottoman immigration procedure. If you're an Ottoman official in charge of evaluating migrants to enter the city (in this case Istanbul), you first must look for their (As migrations, with some exception must be taken in group) surety (kefālet). As i elaborated in that link, migrants most of the time must have a person they can trust or guild in Istanbul in order to migrate, and government officials received orders to register immigrants with their guarantors and to expel anyone who lacked one. However occasionally people with legitimate surety were expelled, like in the era of Selim III and Mahmud II.

Sometimes person/persons might want to visit Istanbul for some time they must apply for a travel permit ( mürūr tezkeresi) which is also your job to provide. The Registers of Important Affairs for Istanbul contain records of numerous instances when travelers were stopped at checkpoints for their papers and had to get permits, though generally merchants and artisans delivering provisions to the city were recorded in registers and exempted from such restrictions though sometimes even they were subject to investigation, as during the unrest of 1740. One such example about issuing a travel permit, in 1742 a certain Derviş Hüseyin from Edirne informed the authorities that he needed to travel to Istanbul and stay there for a few days to settle some personal business. He then applied for a travel permit to enter the city with a woman in his company. People who came to settle personal business and petitioners traveling to submit their case to the Imperial Council had to receive a certificate of permission from their local judge, and delegations from the provinces were to be restricted to one or two people. However, even with regulations, immigrants often eluded controls, in part due to the neglect of state officials, but more importantly, because the city’s dynamics generated a constant demand for their services.

Sources:

Artisans of Empire, Crafts and Craftsmen Under the Ottomans by Suraiya Faroqhi

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