All members of the House of Osman were ordered to leave Turkey by a decree on 1st November 1922. The last Sultan, VI Mehmed Vahdeddin left the country aboard a British warship, HMS Malaya on 17th November 1922. His first stop was Malta, then a British colony. He spent some time in Mecca, no longer part of the Ottoman Empire, and ultimately settled in San Remo, Kingdom of Italy. He died there in 1926; he is buried in Damascus (it was an Ottoman land and he, for obvious reasons, could not hope to be buried in Turkey).

The last caliph, also belonging to the dynasty, left Turkey following the decision of the Ankara government on 3rd March 1924 to abolish the Caliphate. He first went to Switzerland and ultimately settled in France. He died in 1944 in Paris.

The remnants of the dynasty often lived peaceful, mostly apolitical lives in Europe and the US. Very early on, the leaders of the dynasty did draw attention to their situation and but nothing of a general anti-Republican campaign emerged. Especially in the later generations. A few generations later, elderly 'Osmanoğlu' figures (that's the surname in Turkish) even remarked Atatürk was right in abolishing the sultanate and so on. That said, of course, there were attempts to do things differently politically. In 1924, during his stay in Switzerland, for instance, Abdülmecid made a call to the Islamic world and declared Ankara's abolition of the Caliphate as unlawful. This did not amount to much ultimately, and soon the Swiss authorities asked him to abstain from such political moves which they deemed created tensions with the Ankara government.

For some decades now, members of the dynasty can obtain Turkish citizenship and live here. There is a sort of generational conflict in the family, among those still related to Turkey anyway. Most elder Osmanoğlus still dislike being drawn into politics. Younger figures like Nilhan Osmanoğlu and Abdülhamid Kayıhan Osmanoğlu are politically much more active; they make anti-Atatürk and/or anti-secularist remarks in public and feverishly support the governing AK Party. But there's by now a new sheriff in town, and frankly, if offered a choice between President Erdogan and any figure from the dynasty, people's choice would be effortless to figure out. To this day, there is still no serious, that is large, organised, and/or popular pro-Sultanate and anti-Republican movement in Turkey.

Source:
Cevdet Küçük, 'Abdülmecid Efendi', TDV İslam Ansiklopedisi, (1988). Available online at (Turkish) https://islamansiklopedisi.org.tr/abdulmecid-efendi (20.05.2020)