I don't think Heavy Cavalry is supposed to be able to swim, so it probably wan not exhaustion...

I will have to give this a go... Wait until a unit is swimming and then send a non-swimming unit to attack them and see what occurs..

Scout: The enemy are swimming the river sir!!
Commander: ah-ha... Bob, take your cavalry down there and kill them all...
Bob: umm... errr... I don't think...
Commander: I don't pay you to think, get going...
Bob: But I don't think I can swim in all this armour!!
Commander: Are you refusing a direct order!?!? I will have your head!!!
Bob: Ah.. well... of course we will attack immediately!!

Bob: Right lads, our target is the men in the water, form line and CHARGE!!!

Splash... gurgle... bubble...

Swimming Barbarian: Well that was odd!!!