... the Muslims and E. Christians? I would probably say the Mongols would ravage everything and leave, the coastal forts being too much bother. Whether or not Latin power would be broken is hard to predict, and I don't know if a Mongol army could even actually reach Egypt, the Sinai not being a terribly good route for a huge horse army.
It's just a footnote, but crossing that desert might not be
too much of a problem; IIRC a Mongol army could travel about a hundred miles a day (if not more), and if that's correct, then they should be able to cross the Sinai in one or two days if they follow the coast. (IIRC it is only about 220 kilometres from Gaza to Damietta, and the Mongols had an outpost in Gaza in OTL)
And along the coast, Mongol scouts and vassals (i.e. the Crusader states and Cilicia) could set up a few outposts/supply camps along the coast between Gaza and Damietta.
The Mongols would certainly need to adapt to the local circumstances, but I'm quite sure that they could do that.
..
As for the fortresses and breaking Latin power; going for the easy targets and not bothering with fortresses was mainly a tactic that the Mongols employed during reconnaissance raids, raids for the purpose of plundering, and the initial stages of an invasion.
After the countryside and weak targets were taken care of, the Mongols would send in a force to take care of large armies and/or heavily fortified cities.
But if the Mongols weren't interested in conquering an area or just felt that it was not the right time for conquest, then they just raided the countryside and avoided the confrontation with the enemy.
And Egypt was rich and definitely worth conquering, and the Mongols certainly intended to conquer Egypt in OTL - so it seems rather likely that the Mongols would make at least a serious attempt to conquer and hold on to Egypt.
Wether a Crusader state in Egypt would resist the Mongols in the first place is also a thing that remains to be seen; Cilicia and Antioch had sided with the Mongols, and a Crusader state in Egypt (which would very likely be weak and somewhat unstable, and quite propably involved in a lingering conflict with a remnant of the Ayyubid Sultanate of Egypt that survives in Upper Egypt, as I can't see the Crusaders take or hold on to
all of Egypt) may just choose to submit to the Mongols, just to avoid a direct invasion.
And if the Crusader state in Egypt would be fighting a losing war againest the Ayyubid (or successor) state in Upper Egypt, then it is not unlikely that the Crusaders might just call in help from the Mongols...
If the Crusaders were able to hold onto Egypt somehow (and presumably Palestine and parts of Syria), I would think liberation would have to wait for the Ottomans.
This is more of a general note rather than a reply specifically to Abdul, but there are a few things that I feel are being overlooked in this debate: when discussing the effects of a Crusader state in Egypt during the period of the Mongol invasions, then the main point is not about what's there, but what
isn't there - namebly the Mamluk sultanate.
For most of its history, the Il-Khanate was held in check by the Mamluks, and conflicts with other enemies (primarily the Golden Horde, but also the Chagatai Khanate and Qaidu Khan) prevented the Il-Khanate from focusing on the Mamluks.
Without the Mamluks, the Il-Khanate lacks one very powerful enemy, and consequently, that leaves a lot more resources for campaigning againest its other enemies, including not only rival Mongol khanates like the Golden Horde and the Chagatai Khanate, but also rebels within the Il-Khanate, such as those stubborn Anatolian Turkish frontier principalities...
...and the main reason why these frontier principalities succeeded in breaking away from the remnants of the Rum Sultanate and Mongol rule IOTL, was that the Mongols had other concerns than enforcing their rule in Anatolia.
In a scenario without the Mamluks to worry about, the Mongols of the Il-Khanate are going to have a lot more time to worry about those rebellious frontier principalities in Anatolia
and they'd be a lot stronger, as the Il-Khanate wouldn't be worn down by it's conflicts with the Mamluks.
And any change in the political situation among the frontier principalities in Anatolia at this point may very well result in the region being rather affected by butterflies, with a good chance that the Ottomans (or at least the Ottoman Empire) are butterflied away alltogether...