@RexSueciae You have a pretty good Ming TL in the works, do you know anything about this subject?
I am actually a direct descendant of
Yang Jisheng, who made his career during this exact period (1550s Ming China).
So the consensus seems to be that IOTL, Altan Khan wasn't much interested in conquering territory or unseating the emperor -- his intent appears to have been raiding the Chinese frontier (as far south as possible, culminating in his raids on Beijing itself) in order to bring the emperor to the negotiating table. And that's exactly what happened -- the court decided that it would be a good idea to officially sponsor markets that would trade with Altan Khan.
Yang Jisheng thought this was a stupid idea and said as much. He was subsequently arrested, demoted, and sent into internal exile (but then quietly returned to office once everyone realized the market plan wasn't really working and that the northerners were still causing trouble -- and then he criticized the corruption of the Grand Secretary, who this time arranged for Yang Jisheng to be executed -- long story).
But why did all that happen? What motivated Altan Khan to take this course, and what motivated the Ming to accept? For the first point, it's important to note that nomadic horse cultures (like the Mongols under Altan Khan, or for that matter the Mongols under Genghis Khan) can get along just fine with what they can raise on the steppe -- from horses and more importantly sheep -- but there's certain goods that a nomadic civilization simply cannot produce and must trade (or raid) in order to get. Obvious examples might include metal tools, armor, and weapons, but maybe also luxuries like fine silk clothing (because people in the past liked to look good just like people in the present). A chief / khan / sovereign who got a lot of stuff from the neighboring agrarian civilizations and could, for example, outfit his personal guard in shiny metal armor would be a man to be respected and feared.
(For an in-depth yet accessible look at how nomadic cultures on the Eurasian steppe did their stuff, I recommend Bret Devereaux's four-part series of blog posts on the subject,
the first of which is linked here. He basically spends his time tearing apart George R. R. Martin's claim to have "based" the Dothraki on the Eurasian steppe cultures and on Plains Indians. Devereaux incidentally explains in detail how those societies
actually worked, which was a fair bit different from GRRM's weird fantasies.) (Please note I don't mean to say that writing in the
fantasy genre is weird, but Devereaux seems to think that GRRM not only doesn't know how people lived on the Eurasian steppe or the Great Plains, but that GRRM doesn't even understand how such a society would function.) (Bret Devereaux's blog is super useful -- I might make a post in the Resources thread one of these days -- I highly recommend his content.)
Back to the point -- it was in Altan Khan's interest to 1) raid China 2) demand benefits for not raiding China (which isn't to say he was precluded from turning around and raiding China anyways, because what the fuck could anyone do to stop him?) (this is part of why Yang Jisheng was skeptical of the horse markets plan). And Ming China was more than willing to give him stuff in the hopes that he would go away.
That may sound odd but it's how the imperial tributary system worked -- neighboring civilizations would send representatives to kowtow before the emperor and maybe bring gifts, and the emperor would accept their submission and send gifts in return. But, of course, there wasn't much that the tributaries could bring that was of value to the emperor, and the emperor's gifts to his tributaries would usually be a lot more valuable than whatever he received. Small wonder that eventually the emperor would restrict the number of times tributary missions would be received -- the tributaries knew quite well that if they kept showing up, they'd keep getting free stuff!
The sources suggest that Altan Khan offered horses to trade, and he might have offered sheep as well or other such resources as he would have had, but that's not as important to Ming China as you might think. I don't expect these were draft animals or European-sized war stallions (to the contrary, the Mongols tended to prefer smaller horses), and although you might have read about the War of the Heavenly Horses, that was back during the Han dynasty -- a millennium and a half in the past! -- ancient history even during the Ming, and the stuff of legend more than history. Horses were nice, yes, but remember, the stuff that China was handing out in return was stuff that the Mongols
could not make by themselves -- at least, not easily. (This was the other part of why Yang Jisheng was skeptical of the horse markets plan.)
So back to your point -- could the Mongols have plausibly taken Beijing with enough shenanigans? Personally, I don't think so. You'd have to put your hands on a leader who is content not only to get cool stuff that he could use to intimidate or conquer his rivals, but a leader who is intending to take and hold territory -- to assault a fortified city -- to do everything that Genghis Khan did, so it's
possible, but I'm not sure how the motivations would arise during this era. You might need a different Altan Khan. I don't think you'd need a different emperor -- during the 1550s, when all of this was happening, the Jiajing Emperor was already neglecting most of his duties in order to pursue alchemical elixirs of immortality (spoiler alert: mercury is actually poisonous) and most daily affairs were in the hands of his Grand Secretary, Yan Song. Yan Song was not necessarily incompetent, but he was corrupt, and he spent a lot of his efforts stamping down on criticism (see also the case of Hai Rui, a rather more famous individual who was lucky enough to escape with his life). So you've already got a weak emperor who's devolved power to a mediocre subordinate -- little different from a regency during an emperor's childhood, I should think -- and nothing bad happened. Well, plenty of bad stuff happened. Nothing existentially bad, though.
Then after the Jiajing Emperor finally died, he was succeeded by his young son the Longqing Emperor, who reigned for a short time and was succeeded by
his young son the Wanli Emperor. And it's fairly well-documented that the Longqing Emperor was little more than a figurehead, or at least greatly uninterested in the business of ruling, while the Wanli Emperor took the throne as a child and had a regency for some time -- and like his father and grandfather, the Wanli Emperor eventually got bored with ruling and more or less abandoned his duties. Through all of this, the empire kept stumbling along -- raids from the north, Japanese pirates off the coast -- until it eventually collapsed. (And it some time to collapse! The Wanli era was more or less the last flowering of the Ming, IOTL, and the cracks didn't really start appearing until after his death.) I have the feeling that it would have taken an actively destructive (or perhaps very unlucky) emperor to bring things down "ahead of schedule," as it were.
But let's think of ways that the Mongols, whether until Altan Khan or a TTL alternative, could have taken Beijing. To start, I would assume that they'd need access to at least some of the war machinery possessed by sedentary agrarian empires. No, I don't just mean siege weapons (although those help!), but basic things like a sufficient number of men -- steppe cultures are not population-dense, which reflects on their style of warfare -- meanwhile, China was
very population-dense at this time. And of course steppe armies did often trounce the larger armies of agrarian peoples (see Devereaux and others on how the
caracole, repeatedly charging and wheeling away while firing arrows, would have been immensely effective) but besieging a city? Apart from the basics of feeding your army (which bedeviled armies whose logistical programs were a lot more developed than "take it from the countryside"), you've got to keep an eye out for a relief force catching you by surprise, or a sally from the defenders, and then there's the problem that cities tend to be logistical centers where the empire keeps a lot of the grain they've been taxing out of the peasant farmers, so you better have brought enough food for your men to outlast them. Did I mention that food is important? Because it's important. That's why storming the city somehow (either going over the walls, or smashing breaches, or persuading someone on the inside to open the gate for you) is the best bet if you can't get them to immediately surrender.
Can the Mongols still do it? I mean, in Chinese history there are
multiple times that a steppe culture stormed in from the north and founded their own dynasty. In the 1550s...hm. We're about a century too early for the Russians to show up (and at those distances there's little they can do, they might fight border skirmishes but they're not about to lend enough aid to topple an empire). I think my best bet for Altan Khan is to co-opt the Jurchens (who, after all, IOTL ended up knocking over the Ming dynasty on their own), since the Jurchens were the sort of semi-settled, partially agrarian people who would be invaluable to the cause. And like the OTL Jurchens, they shouldn't stop with just one group -- they should keep going, get Joseon on their side by force or diplomacy, get some Chinese on their side as well -- when the Jurchens founded the Qing dynasty, it was with the support of quite a few Ming defectors who threw their lot in with them.
It took Nurhaci until 1616 to unify the Jurchens under his personal rule. Beijing didn't fall to the Qing until 1644 (okay, to be fair, they took it more or less unopposed after it had already been sacked by rebels). The last Ming emperor-in-exile wasn't captured until 1662. The last Ming loyalists on Taiwan didn't surrender until 1683.
If Altan Khan got the idea to take China -- actually take it for his own, not just pillage it even more intensely -- he might be able to start the preparations, but it would likely be his son or even grandson to finish the job, as what happened with the Qing. (For that matter, as what happened with Genghis Khan and his wars of conquest!) My best bet? If you want Altan Khan to take Beijing, schedule your POD a generation or so earlier.