They definitely get a speed increase. When I tried out the Turks on VH battles the enemy bodyguard was overtaking my own on the map
I actually don't really like this feature so in future SS hotseats I'll likely be setting the battles to H instead. On the plus side it means the days of using a few horse to take a fully defended castle are at least partly over. We'll see what happens when my Mongol HA armies come up against someone they have to actually fight.
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