The Mongols quickly shattered into competing factions. The Mongols that threatened th Mamluks and the Mongols who controlled the actual steppes were different factions. The latter was the "Golden Horde", the ruler of which had converted to Islam. The Mamluk ruler (Baybars) befriended him and and thus secured a ready supply of new slave recruits from the steppes.

The transport of the slaves was done by ship, so the Mongols of the Middle East could not really interfere. To this end, he also befriended the Byzantenes and the Genoese, both of whom were not overly fond of Venice, which at the time was the primary trading partner of the Crusader States.

Source: Asbridge's "The Crusades"