Ok, so here goes my opinion on this.
The lockdown is, at first glance, very annoying to deal with because the whole point of the position is to really immobilize your leg and keep you stuck. It accomplishes this very well. What it also does however, which is bad for the person on bottom, is that it really immobilizes their hips as well. You commit both your legs to keep him trapped and therefore have no free leg to either bridge or shrimp with. This is usually what you can use against them.
First things first, you have to think about stabilization and prevention before you think about passing. Learn to make yourself comfortable in that position and make your opponent uncomfortable. Ask yourself; what can he do to me from this position, what do I need to watch out for? The main thing your opponent will be looking for is at least one underhook, and often both underhooks. He can do this by pummeling and also by using a technique called the "whip down" (Eddie Bravo/10th planet name) to use the lockdown to rock your base enough for you to post a hand, thus enabling him to get an underhook and get to his side, where the lockdown really begins to be dangerous.
Goal number one? Don't give him the underhook and don't let him get up to his side. Easier said than done when someone is good at it, but let me give you a few tips.
Don't try to force conventional half guard passing on the lockdown. It's built and designed to combat the traditional way of half guard passing, and you will often get countered. When I play lockdown (used to a lot before, play it a lot less now) I really love it when people try to "smash me" with the traditional head and underhook grip. Even when they are a lot heavier. Since I do only no-gi (luta livre), it's sweaty enough where I can almost always get the underhook aftar a brief stuggle. Don't let him do it. Instead, flow into passes that completely shut down the underhook and prevent him from using the lockdown the way it is supposed to be used (underhook + side)
There are a lot of passes which negate these two things. I could suggest not even letting him get into the half guard, but you'd probably tell me to shut the fuck up and keep to the question of actually fighting the lockdown. So be it !
The first thing you have to get a hang of is this one. People call it a lot of names. The hip switch pass, twister pass, reverse sit pass etc. It's a favorite of Demian Maia. And Demian Maia is awesome. So really study this one:
Lovato goes into great detail so make sure you are paying attention.
If you like to really get close when passing, squeeze a lot and keep constant pressure, you could try this, even though it isn't my go to pass against the lockdown, it works really well if you get good at it:
The idea here is simple. Get low a low enough grip on his butt/hips (like a double leg somewhat) and sprawl really hard. This breaks the lockdown apart because it puts a lot of pressure on his feet and will force him to open it as a natural reaction, especially against people that are not flexible. From there I like to do a back-step and pass to side control. If he pushes your head down so you can't pass, just switch to the other side. No problem.
With these two clips, you already have a lot of homework. Work on preventing him from getting the underhook and from grabbing behind your leg. This is what he needs for the electric chair, which is a high % sweep from the lockdown. Prevent those two things, and his lockdown will be useless. If he's smart, he'll disentangle himself and play something else. If he's not, he's really open to any of these two passes. If he is able to grab behing your leg, sprawl immediately and reset. If he manages to get an underhook, do a very strong whizzer (I like to throw an overhook where I"m actually putting my fist on the floor. So imagine it like this: he get's the underhook and manages to get on his side. Don't waste time, just put your fist on the floor right next to his hips (this will make the overhook very strong and will put pressure on his arm) and then simply use your free hand to push his face away. Be rough, don't be gentle when you are defending a sweep. Once you push his face away, his underhook will be useless and open. Proceed to pummel your arm back into your own underhook and continue passing.
One last pass, the one I use the most these days is the near-side underhook one.
This one. Very good explanation.
So now you are wondering; how do I do this versus the lockdown?
I can't find a clip of this one youtube, so I'll try my best to explain it by text.
You do everthing like you would in this pass, but you have to untagle his lockdown first. So let's say he has your right leg leg in the lockdown. Do everything like he says in this clip, and once you establish the nearside underhook and are about to put your head on the other side of his head, use your left leg to untagle the lockdown. Find a partner and try it out. Just take your left leg and try to pry the lockdown open. It will feel natural and you will actually establish a grapevine on his right leg. Use this to eliminate his bridge, and just continue passing to mount like you would normally.
Hope this helps, if you need anything else explained, just ask.
Cheers and good luck in training