Well, the caliph of Islam was never in Mecca. Ever. The original capital of the Islamic state was Madinah. It moved to Kufa during the time of the fourth caliph Ali since that is where his main supporters were. It shifted to Damascus after him since the next caliph was the former governor of the province of Sham which had its capital in Damascus. It then moved to Baghdad with the shift of the caliphate to the Abbassids (whose support came from the east) and then to Cairo, Egypt with the sack of Baghdad by the Mongols. With the formal transfer of the caliphate to the Ottomans, Istanbul became the capital of the caliphate and remained so until its fall in 1924.

Because the Caliph was also the civil and military leader of the Islamic Empire for much of its history and if he stayed in Mecca would have been hopelessly disconnected from what was going on in the main population areas (Syria, Mesopotamia, Egypt, Persia) or the border with the Byzantines in Anatolia. The Abbasid Caliph ended up in Baghdad for several centuries because that was the heart of their empire.