Yeah Alex gets WAY complicated when it comes to orders of battle but the basic principles seem to hold in essence for Unit types as far as the function goes meaning "Smoke without fire" results from unsupported cavalry assaults all of it flowing from rule numero uno "It takes boots to hold ground" and everything else is in support of this.
That said the innate advantages of mounted troops are mobility and shock value and We're back to Units and the summing of parts part in large part as cavalry is all about the disordering of Units in either the charge that generally resulted in the running down of a broken mob the same as a flanking maneuver would and often cavalry was reserved for specifically this purpose so as not to allow a broken Unit the space and time to rally back to order
Mobility does have an interesting wrinkle to it tactically being a ground over time consideration such that Prince Rupert's charge situations develop in relation to the ever shifting give and take between technology favoring the attack or the defense at a given time mother necessity being a member of the bitch goddess pantheon and whatnot
Rome's relation with martial cavalry is particularly notable I think as with it or on it culture that stylized its principal adversaries kit in gladiatorial types in the american soccer of its sport entertainment with cavalary by the first century BCE existing as mixed cohorts
I'd hazard a guess that the hitch a ride bit relates to a Napoleon's flying cannon they aren't supposed to be able to get all the way over there type thing