When we say a unit must be within Rapid Movement of another unit to close assault it, we don't mean that it must MAKE a Rapid Move, we just mean that the unit must be within movement range (Rapid Move being the maximum distance a unit can move). There's no problem with a unit assaulting another unit within LESS than Rapid Movement distance or using a Tactical Move. We're not modeling the impact of armored horsemen, so we're not worried about whether or not there's space for an assaulting unit to get up to a gallop.

Hope this helps!