Maybe handle it as part of the fiction? Most of the Dark Ages sources that Fallen Empires draws on don't really focus on killing horses, although they do get killed now and then. 
Also, need to distinguish between mounted gangs that get off their horses to fight battles, which is most common, and actual cavalry.

EDIT: Hidden in the description for Sword-master is a description of the benefits of fighting from horseback: +1 Harm, +1 armor against non-mounted enemies, mounted enemies get no advantage against you. 
I would think those same benefits would apply to mounted gangs fighting against gangs on foot, but who knows? If you do go with this rule, it would be a good idea for it not to apply, or not apply in full, to infantry in defensive positions or shield-walls. 
I would probably treat mounted as a tag that can be applied to gangs, and which can be lost if somebody steals their horses or, for example, if they're besieged and forced to eat them all to survive.
                                                                                 

It occurs to me that this is sort of how I ran my most successful AW game -- it was in a desert loosely based on an old Route 66 map, and I planted communities  semi-randomly up and down the roads, wherever there was a water source and a place to grow food.  The roads are still pretty good, and since it's Apocalypse World, there's still plenty of gasoline.  So conflict went through lots of little towns like beads on a string, and I could Announce Badness with the big bad Warlord by counting off the shithole towns he'd taken as his forces marched down Route 66. 

A setup like that, with a network of smaller communities linked by roads, would make a perfect setting for this kind of game.

About to start my game and very, very pumped to see how Rights work in practice.
I feel like a lot of our discussion in this forum gets hung up in different conceptions of "right", mostly the modern idea of inalienable human rights versus the narrower concept of legal rights to something. Looking at the Magna Carta, I'm seeing Rights as working more in the second sense, as a way of keeping armed struggle from breaking out by respecting some minimal, essential rights belonging to powerful people. 

Yeah, I like that. Sort of a crossbreed of Hardholder and Operator, where the gigs are whole towns. 
 I think it could work. You know how most AW games start with a sketch map: here's your town, up north are some cannibal bastards, to the east are a bunch of disease-ridden farmers, and here's ruined city full of bandits and stuff.  In a game with a Tyrant, you start of with your town plus a couple of towns, and each of them has their own crap to deal with and problematic neighbors. 

And then you've already defined the setting more, and it will be easy-peasy to make NPC-PC-NPC triangles with the Tyrant, because the Operator's got a gig running back and forth to the other holdings, and the Gunlugger just shot up one of the little holdings because they pissed him off, and maybe the Savvyhead found a cache of ancient disk drives in another holding and is trying to keep it quiet so she doesn't have to share it with the Tyrant.  I would totally play this!

I've been thinking about creating a Warlord Playbook that would be "man with an army" the way Driver is "man with a car" or Hocus is "man with a cult" -- an actual standing army, with sergeants and rules and marching order would be an unusual and fragile thing in Apocalypse World!

I think there are two, maybe three things that the Warlord or Emperor has to be able to do:
1) Conquer new lands -- maybe a modified version of the Hardholder's military move, you'd probably play this out but eventually you need to leave a stooge in charge and go home to check on the treasury.
2) Manage your stooges -- maybe a modified version of the Operator's gig -- create an NPC lieutenant that you're putting in charge of Joe's Town, roll every session to see if they're still scared of you and doing their jobs.  It could be one of those Moves where on 10+ you choose 3, on 7-9 you choose 1, with results like "the stooge remains loyal, the stooge is keeping the territory pacified, the stooge isn't stealing from the locals and keeping it for himself". 
3) Hold power -- an Emperor needs secret police of some sort, to protect him from disaffected subjects and from greedy officials who are skimming from the top, and especially from generals who decide they want to be Emperor, too. 

If you look at Roman history, you see a lot of Caesars blowing these rolls and getting their legions wiped out, or having whole provinces rise up in rebellion, or getting banked by a fresh-faced general riding the high of victory. 

I could totally see an Apocalypse World game in this kind of setting, but it might be a little mismatched in terms of scale -- one PC is trying to conquer the whole Napa valley for his California Empire, while another, back in the ruins of Sacramento, just wants to fix up his movie theater and sell real popcorn with fake butter. 

I am not an authority, but it seems clear to me that Rights are real things IC in the game setting. If you have the right to claim.a sacrifice every year, that's your legal right and denying it might be a big deal. If you asked a neutral third party in the setting, like a wise old NPC, he/she would say the character has that right.  The thing is that the AWDA setting doesn't have lawyers where you can sue somebody, you have to start a fight, or plead your case to another strongman who might intervene, or pray for the gods to avenge the wrong done you.  And all of those are real options.

But I'm pretty sure that Denied Rights have nothing to do with OOC interactions or the player getting pissed off. That's a whole different thing.

I think this is a very interesting approach.  It's quite logical that a Hardholder might want to make the jump to warlord.

Another approach, if you wanted to keep things small-scale and gritty, is to make the player define, repeatedly, what they mean by "conquest" and "control."  It's one thing to send a militia of guys with rifles into the next town to shoot the mayor, and another to have that other town remain calm and provide benefits back to your holding.  Who's left in charge of the town -- a local guy who promises to be loyal to the Hardholder, or one of the Hardholder's men, maybe with his own little army to keep him in control? 

Maybe the Hardholder will reinvent the feudal system, start disarming the populace and build a fortress for himself and his guys. 

If he succeeds, then run a follow-up game set in one of the towns that the Hardholder PC conquered and pacified.

I've been thinking lately of a couple of  interesting ones:

-One is inspired by New Wave Science Fiction -- the characters find themselves in a biological landscape, everything around them is some kind of tissue or organ, only impossibly large and baroque.  There's a huge 3D space of twisty little passages with a constant wind blowing in and out to represent the lungs, for example; somewhere else, there's an endless forest of trees crackling with lightning, representing the brain.  The rivers are full of giant blood cells and deadly antibodies.  Anyone who falls into the heart is liable to be washed away and find themselves in desolate landscapes without any human presence at all. I'd run this in a heartbeat, but I can't figure out where to take it from there -- whose body is it, and what are the PC's doing there? 

--Another is inspired by the cool Dungeon World setting, Inverse World, where everybody lives on floating rocks above a golden sun, and the rain comes up from below, then drips back again.  Inverse World is a setting where things just work like that, but I really like the idea of playing in a setting where the apocalypse was gravity suddenly reversing -- most of the people and animals were outside, behaving normally when gravity failed and they took the long fall.  Some lucky people were inside or on the underside of rocks or in airships or could fly, and so some small fraction of the population survived to colonize what were formerly the barren undersides of the worlds, eternally in darkness since the sun is now below them.  (This is actually part of the Inverted World setting -- farmers grow crops that hang down from the underside of floating rocks, since that's the only part of the land that has both sunlight and rain.  Being a farmer is dangerous work.)

Some of the planetoids and floating islands themselves broke up under the strain, shattering into dust and dangerous flying meteors.  Hell, maybe most of civilization once lived on some great floating continents, most of which have now shattered.  But if you've got ship, you could cruise along the underside of a dead land, cautiously climbing out into upside-down towers and buildings that still somehow cling to bedrock, looking for treasure and daring the wrath of the insane, cannabilistic apes who have colonized the inverted ruins....
I'd probably have the PCs come from either a dying little rock in the fringes, or an ancient airship left over from the Great Breakdown, constantly in need of repairs, constantly moving through the wreckage like a steampunk version of Battlestar Galactica.

By the way, I have a severe phobia of falling, so this is the scariest possible Apocalypse for me... your mileage may vary.