It seems to kick in from the 1st upgrade.

For pure desert fighting I find 1 armour upgrade is usually ok. Units with 2up on armour tire faster; even jinettes.

Units classed as "armoured" tire faster, period. Feudal Sergeants best spear; gallows/highlanders best sword; vanilla archers best missile. Anything above "light" horse will fatigue quite quickly.

Morale upgrades much more valuable than armour in the desert.