Yes reinforcments do come on already tired, however that is better than tottaly exhausted and unable to recover. The best way I have found to use heavily armored foot soilders in the desert is to bring lots of them and as they tire rotate them off the field. Also do not let them chase routers, use lighter troops for that.
As far as heavily armored cavalry, they are just too slow and tire too quickly to be of any use.
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