As far as I can tell you can, but I'm curious if the following examples hold true (or if I'm completely wrong in the first place).
(a). You're riding a horse out of combat and decide to "shove" it into the ground. Since the horse probably wasn't expecting you to be a source of danger, I'm thinking it's safe to assume that it's surprised, thus can't act on the first round of combat that you just initiated. You attempt the shove and succeed, forcing the horse prone (and you're probably thrown to the ground as well), effectively stopping it's movement until it can act next.
First, if you do this, you're a jerk. What did the horse ever do to you?
While I'm not sure that I'd consider it a "shove" by the dictionary definition, I might actually allow the use of the Shove attack to attempt what you're describing. I'd do it with disadvantage, for sure, because 1) you don't have any actual leverage other than your own body weight, 2) I've ridden horses and can't picture a way to knock one down without doing more damage to myself than to the horse, and 3) it's got four legs pretty much exactly where it wants them to be to resist you. Actually, I'd probably give the rider disadvantage because of #1 and the horse advantage because of #3 and not feel guilty about stacking the deck because of #2.
As far as surprise goes, I'm kinda serious about the jerk thing. If you're on the horse and safely riding it, why would you even want to trip it up? Maybe if you see an obstacle that it doesn't, but that doesn't really give you a reason to care about anything beyond the first attack. In that case, sure, why not? If you're actually intending on killing/disabling the horse, how did you get on the horse's back, in the first place in a non-threatening way?
(b). What if you did it while riding on the back of a Wyvern? Would it immediately fall to the ground prone? Are there rules I've missed somewhere stating how far you can fall in a round, or is it just assumed that the Wyvern has no chance to recover before hitting the ground?
Numbers to note: If I'm doing the math right, a critter can be assumed to fall about 575 feet in the first round, assuming complete loss of lift and ability to act. Anything above that height, we'll just assume recovers safely, but needs to change it's underwear. Also, terminal velocity (about 200 ft/sec, allowing for some drag) kicks in at about the six-second mark, so, we can just assume anything after that is a stain.
I'm not aware of specific rules for knocking a flying creature prone w/o also disabling them. So I'm going to assume that, barring unconsciousness, being "knocked prone" while in flight really just means disrupting the creatures flight, momentarily. Why? A combination of common sense and game balance: I can't see anything short of a full grapple completely destroying the wyvern's ability to fly for long enough to totally fall and it would make a Shove attack too potent and the only sensible option in aerial combat. Since getting up from prone costs half movement, so successfully "shoving prone" a flying creature would just cost it half its movement with no further effect. If you want to change its elevation, use the Shove attack directed downwards.
(c). What if you grappled instead of shoved? Are you literally pinning a horse to the ground after putting it into an Equine Full Nelson? Is the Wyvern totally screwed as you both plummet to your deaths?
Looking at the Grappled condition, the only real effect is to reduce the critter's speed to 0. So, go ahead and make the check. I might give the horse's rider disadvantage, just because that's a somewhat sub-optimal point of leverage, but it also makes sense that the rider is going to swing in front/underneath the horse, too. So maybe give the rider the choice of either taking disadvantage or do a rather uncomfortable dismount as part of the grapple -- being left behind, if he fails.
The wyvern might be similar, but doesn't have the ground to bash the rider with. Cinematically, I don't like the idea of a failed grapple resulting in the attacker going into free fall, solo. So, go for it. If you succeed, you're both in it, together, and share the date with the dirt.
Now, two flying creatures grappling might be pretty cool, as each tries to trap the other in free fall while pulling themselves out, at the last second. YMMV.
Anyway, am I missing something or would that stuff really play out the way it seems?
Depends on your goals. I hate rules for the sake of rules, in RPGs. Story first, with the rules there to add just enough structure. If common sense says you can or can't do something, then common sense prevails. Ditto for genre conventions and narrative flow.