My historical knowledge regarding Ali pasa Rizvanbegovic is not that great but what I do know is that he was responsible for Hercegovina becoming an independent eyalet within the Ottoman Empire (separate from Bosnia). This was in 1833. The Rizvanbegovic family was from Stolac in Hercegovina but owned much land throughout Bosnia-Hercegovina, including land that my ancestors lived on near Duvno or today's Tomislavgrad.
In 1878, Austria-Hungary annexed Bosnia-Hercegovina to itself. This most likely upset many Muslims who did not want to live under Catholic rule. Many chose to emigrate during this time as a result. Today, their descendants can be found in old parts of the Ottoman Empire, modern day places like Turkey and Palestine.
Your surname Bushnaq is probably derived from the word Bosnjak meaning "person from Bosnia". This word refers to anyone living in Bosnia regardless of religion. However, today the name usually refers to Bosnian people of Muslim faith. It is also a common surname. Many people in both Croatia and Bosnia-Hercegovina carry that surname.