Basically, the travel speed was the same for everybody, only the comfort differed. The most comfortable possibility was an ox-drawn cart called goshoguruma, reserved only for the highest aristocrats and the Imperial family. However, the speed was also the speed of a walking man, because as you can see in this photo taken at the Jidai Matsuri parade in Kyoto, the goshoguruma drivers were walking beside the cart.
EXIF info:
Nikon D700
Lens: VR 70-300mm F/4.5-5.6G
Focal Length: 200mm
Aperture: F/5
Shutter Speed: 1/200s
ISO Sensitivity: ISO 500