I don't mean this as any offense, but entire styles of wrestling have been created to address this very issue. It is the equivalent of asking, "once the fight goes to the ground, can you give me some tips or details on how to get an advantage?" Your best bet is to pursue Greco-Roman wrestling, Sumo, or if in the Gi, Sambo, if you are a huge fan of this position.
With that said, knowing that it is an artform in and unto itself, I'll give you a few tips that will hopefully help.
So, I'm going to assume you start from an over under position, though I don't really believe this is natural. Generally, you're only going to have one side engaged in an over/under, with the other hand free to post on the ground for a scissor, scoop an ankle for an ankle pick, snap down to a front headlock, or what have you. To use an analogy, it's why some Filipino Martial Artists actually prefer having one hand empty while stick/knife fighting; the utility factor of the empty, free hand cannot be overstated. However, this may be part of a drill, or you may be wrestling under a particular ruleset, that requires you to go to the over/under position. Or you end up there organically during the match.
First, the purpose of pummeling is to get double unders. Double unders is a game ender. It means you should win, and if you don't take the other person down, it's not because they are good, but because you screwed up. The #1 super easy best over under finish? The suplex. It creates fear in the heart of men, looks cool, and is incredibly functional. Some people will substitute the double under suplex with the tried-and-true BJJ squat-down-on-one-leg-and-trip-them-over-your-outstretched-leg, but this is an inferior version of the technique, because your opponent can twist in such a fashion to land on top of you, negating your takedown and extreme advantage you had. A baby suplex is to simply squat your butt to the ground first, and do an extremely low altitude suplex. No injuries, and you land on top! Hurray! Alternatively, if you are dead set on not being cool and suplexing, you can drop down to a single.
Now, you know you should be rushing for double unders as soon as you can, but your opponent knows that too! So, many times, you end up in interstitial positions where you need to tip the balance in your favor so you can take the other person down. The two important factors here are head angle, and wrist control.
Why?
Head angle, which is defined as having your forehead on the side of your opponents face (or if you prefer another example, if you're both cyclops from x-men, you want to be zapping him without being zapped in return), is an incredible advantage to have. It pushes at the far end of the lever that is your opponents body, keeps them higher than you, and can allow you to force someone down with headbutt power alone. If you can't get double unders, the next best thing is head control and an underhook. With this you can ankle pick (easy, low amplitude) or belly to belly suplex (cool, high amplitude).
Next is wrist control. Rather than doing a full on over/under, I'll retract the overhooking arm to get a grip on my opponent's wrist. Your opponent, unless you are totally and completely outclassed, cannot take you down with one hand. They are structurally weak, and you can exploit the controlled hand to neutralize their takedown attempts. So wrist control gives you safety; you know as long as his wrist is held, you are safe. If his wrist is loose, go on high alert! Having an underhook, head control, and wrist control means you are doing pretty damn good, and should be able to take the guy down.
But let's say you're missing one of these three components (underhook, head control, or wrist control). What do you do?
Not having the underhook, but having head control and wrist control, is still a good position to be in. But how, you may ask yourself, can you have head control with an overhook? Aren't they under you, what with the underhook and all? First, make sure you have a real overhook. This means pressuring HARD into the shoulder, as if you were trying to lay all of your bodyweight forward through their shoulder. This is incredibly uncomfortable, and gives you control over their joint, as opposed to the reverse. Now, they're going to be pressuring hard into you with their underhook, to try and take control of your shoulder joint, and such the game is played. If you have an overhook, but you also have head and wrist control, you can still takedown (ankle pick, for instance). In fact, some people prefer this position (generally people with bad shoulders).
What happens if you lose 2 out of 3? So, for instance, your opponent has the underhook and head positioning, but you have wrist control? First, hang on to that wrist, because you're one more mistake away from being suplexed. The best way to reset the position, to give you another bite at the apple, is kind of a fake uchimata; you kick back and up, hard, such that your opponent (and yourself) are headed straight towards the ground. Their head is thrown forward through gravity. This gives you the opportunity to regain head position (and if you do, now all their weight is on one leg, making the ankle pick extremely easy; you essentially just drive forward and finish at that point). If you have head control, but they have your wrist and the underhook, you can try and use your far knee (not the knee near them) to strip the grip and re-enter the handfight.
Finally, let's say you've really screwed the pooch. Your opponent has double unders on you. You don't want to be suplexed, right? No one does, because it's humiliating; it's something you do to other people, not something that is done to you! Immediately put your hands on the floor, do a forward roll, grab for one of their legs, and try to enter some kind of guard. There is a series of leg attacks you could do, such as viktor rolls, a roll into x-guard, or even playing an inverted guard into DLR, but you need your hands on the mat asap, to start turning over, and looking for a leg to hold on to, to keep you landborne instead of airborne.
This is a mini-game plan or roadmap you can follow to make sense of the clinch, but if you're looking specifically for greco-roman takedowns, you're into the fundamentals of any takedown art; off balancing your opponent, making one leg light and the other leg heavy, and getting your hips below your opponent. Just playing around with these concepts, you'll accidentally take other low level people down, and get an idea of what things are supposed to feel like.
For now, even if you're getting smashed, record it all. See what your opponent is doing to off balance you. See how they throw you. If it's part of an overall grappling spar, then use the methods I mentioned above, and look for the hierarchy of grips. If it's part of a drill, just try to learn as best you can. Move people around, feel how they move, and try to create attacks or throws based on that feeling (i.e., if someone is bending forward at the waist, they've already adopted the position of being hip thrown, so throw them! Or, likewise, if they're bending back, they've done half the work of being backheel tripped, so trip them!).
tl;dr:
"How do standup clinch?" isn't an easy question to answer.
