Slow work, stop & back at the barrel. I only do that when they don't rate though. I teach mine rate before they see barrels. To signal mine to rate, I sit and squeeze my legs  at the same time. Start out by using a voice command with it, I say Easy. I call it my easy button. Whatever speed I'm going I sit, squeeze, say easy all consecutively. I personally don't see rate as a slowing, I see it as get your ass under yourself. The sit and squeeze should signal them to do so. Long trot the sides of the arena, at various intervals ask for the rate. If they ignore you and give you the finger, stop & back. Eventually mine get to where I can just sit & squeeze, don't have to say easy. 

I agree with stopping and backing. But I would also turn the barrel twice when I’m doing slow work at a walk. Especially since he’s bowing off.

The key is teaching him to react to your shift in weight (add a verbal whoa if thats your thing). Off pattern starting at a walk you overexaggerate a lean over the horn in a hurry/run position. Sit deep and firm and ask him to stop. Do this at all 3 gaits until when you shift to sit he reacts by slowing and stopping. This accomplishes a cpl things, first it teachs him weight shift as a rate cue (which is good because it mimics body position during a run) and helps keep you from having to handle him much during a run. Everytime you handle their face during a run you loose time. 

Next after he is a pro at the above set your barrel pattern about 30 feet apart. A short pattern is great for keeping their mind on turning and not running. During your pattern, lean over the horn like you are asking him to run. At different spots sit and ask him to stop. Stopping at Different spots keeps him listening to you. Do this at different gaits. Never repeat the cue in the same spot over and over, it doesnt keep him really listening to you. You can shift your weight in the middle of the barrels, at your rate point, when you leave a turn... keep him listening. You can do this on a larger pattern, alternate sizes as he learns to listen.

i also do a lot of backing. Not tugging on their face or using your legs (like a pleasure rider might) but backing off my hands only. The goal is to get them moving backward off hands but keep it super light. They should learn to back with very little pressure, without getting bound up or over breaking thru their neck. No chins to the chest, thats counterproductive. 

Ok... I kind of do both.  

Full Disclaimer... I get colts. I do not train my horses; nor do I train/start them on barrels. I get them when they are running or starting to.

So that being said... I do both. If they are early in showing or have been shown, they can lose their way at times.

Sit at the first, stop, back up, repeat if necessary.

And... I also and/or then do, the body position thing... mine and their position.

meaning, if I sit: they rate. And if I raise my butt, then they move forward.

So, it is a matter of figuring it out. Hope that makes sense, if not just ask.