The Abbasid Caliphate by the time of the Mongol conquests was already effectively a puppet of the Turkic sultans and had been for several centuries.

As such the destruction of the Abbasid Caliphate, such as it was, is solely the responsibility of the Mongols who carried out the killing of the Caliph, but the end of Caliphal power had been carried out long before.

There was technically a branch that continued in Cairo but it's almost completely ignored. As an institution the Caliphate was of course restored under the Ottomans but as an office of Islamic authority it bore no resemblance to the Caliphate of, say, Harun al-Rashid.

You may want to specify that question some. Do you mean a specific caliphate, or do you mean the concept of a caliphate as a realistic political entity?