If you want to rationalize it, Yes, that is why they get tired, but in terms of game mechanics the troops that have heavy armor (3 or more to start with iirc) have a sand penalty.

Observation: Playing Antioch campaign i was sieging a rebel settlement, my troops, mostly Edessa Sergeants (or whatever the orange mace-wielding guys are called) and Hospitaller Spearmen, became exhausted after fighting at the walls and gates. I let them rest to gain some stamina before taking the city square, but even after they were standing still for about 20 minutes they had not gained anything, they were still exhausted. Since the timer was running out I was forced to charge in and take the city with less than 2 minutes remaining. This could perhaps be a change for Kingdom: the 'Frankish' soldiers seem to not regain their stamina as fast as they used to in the desert.

One of many minor changes that make a big difference. I'm not saying its bad. Now I always keep 2-3 units back from the fighting so that they stay fresh and can easily finish off the remnants of the enemy.