The trail seems to just go dry on them. It doesn't seem like the last caliph was killed and surely after centuries of relative security in Egypt the Abbasid family would have many living members. Did no one keep tabs on this ancient and prestigious family? Muslims seem to have been pretty good with family trees so I'd be surprised if the Abbasids just vanish into thin air after 16th century.
I don't know the answer to your question, but I just want to point out that the Abbasids did not actually surrender the Caliphate to the Ottomans: from Caroline Finkel, Osman's Dream (2005), 111.
"The title of caliph had, for instance, been used on occasion by Ottoman sultans since Murad II, but in a rhetorical sense rather than as a straightforward political-legal assertion of sovereignty over the Muslim community. Selim certainly made no claims to exercise what remained of the sacred authority of this office; the last caliph, al-Mutawakkil, was sent into exile in Istanbul, where he stayed until the reign of Selim's son Süleyman. As time went by, the issue of the caliphate came to intrigue Ottoman intellectuals, but stories that there had been an official transfer of the office to Selim when he conquered Cairo did not begin to circulate until the eighteenth century, when it was necessary to counter Russian claims to protect Ottoman Christians with claims to Ottoman spiritual authority over Russian Muslims."