Fatamid heavy cavalry had little issue challenging the Byzantine Cataphracts on the battlefield.
We're talking horse archers vs other cavalry here.
It's not as if the Mamluks had an issue fighting the Mongols- Ain Jalut wasn't a fluke.
Historians would beg to differ:
1. Mamluks used hit-and-run and feint retreats - mongol usual tactics, mongols apparently didn't (possibly because approximately half of the force was auxiliaries, not used to such warfare, such as Cilician and Georgian heavy cavalry and local recruits/mercenaries).
2. Qutuz, the mongol commander, made a grave mistake and ran into ambush.
And even then, the battle didn't decisively go into Mamluk favor for quite some time.
Their heavy cavalry-Light cavalry-shock troop combination was specifically created to combat the Mongols and the il-Khans, and the armies of al-Kamil and as-Saleh differed little from those of Baibars or Qalawun.
Mamluk weaponry and fighting style was borne out of their ancestral lands - Russian steppe (IIRC they are cuman origin), so it was in a way similar to the Mongol, but in general should still have been inferior as even a small mongol
scout force of two tumens (led by Subedei and Jebe) had little difficulty dispersing their armies, when they arrived at the cuman homelands.
It is unreasonable to expect that Mamluks
tailored their army composition to meet mongols as Ilkhanate appeared on horizon (Iran) in 1256, while the Battle of Ain Jalut happened in 1260.
They might have made their troops have more armor, or have some improvements in general quality of equipment, but in the end the fact that they were engaging a small force, the cunning of their commander and the lack of the same on the enemy commander's part were the decisive factors in the battle.
It is not possible to change the "troop type" on a whim, even today giving an infantry division tanks, they don't automatically become tank division - it takes a whole lot of training to make them such.
In the past "troop type" was basically representative of your social status, so apart from minor elevations (extra armor, etc) and tactical moves (dismounting of mounted troops), you couldn't expect them to change easily.
The mongols never made a serious invasion attempt in Mamluk lands.
Heck, even the invasion of Poland was a diversionary raid, yet it managed to devastate the country.
As I said, the main problem of mongols was their lack of political organization - they returned for kurultai each time the Great Khan died and in the later years they were unable to assemble as large forces because of their inheritance laws splitting their forces and the resulting infighting.
The turning to kurultai part saved Europe (1241 - Ogedei's death) and Mamluks (1259 - Mongke's death), after that the Great Khan started focusing on China (Kubilai Khan) and the West and Middle East were thus saved.
Militarily they were pretty much unstoppable once they focused to defeat/conquer someone.
This was a result of more than one factor though:
- training and general quality (incl. material quality of equipment) of troops
- organization and discipline of troops
- experience from constant fighting or hunting
- excellent commanders in large part due to meritocracy
- unseen mobility
Note that numerical superiority was almost never the case - most of the times the size of armies was either similar or in favor of the enemy.
Even more so, when taking into account that often large part of the army was composed of other steppe auxiliaries which were on average lower quality troops.