Andrew Smith is a BJJ black belt based out of Richmond, Virginia. He is one of the owners of Revolution BJJ.
Butterfly guard basics will take you a long way, and once you start playing the overhook game, you have a powerful and comprehensive butterfly guard that will annoy and perplex your partners. However, there's another really important direction (and distance) from which you can attack: the arm drag. We've covered arm drags from closed guard, but butterfly is fundamentally different, and readily accessible for you. Let's take a look at a very clean, simple attack to begin with, and then discuss some options when that approach isn't working for you.
Start by finding the right range with your partner. Make sure your butterfly guard is comfortable when you're sitting up, and you should be able to reach out for their wrist without exposing yourself too much or having to go out of your way. Reach out with your left hand and grab their right wrist. "Grab my arm. The other arm. MY other arm!" Next, use your right hand to cup behind your partner's triceps area, creating an opportunity to pull. While you do this initial movement, it's also important that your left leg doesn't get in your own way, so move it out of the way so that you will have an easy path to the back. You can augment the leverage you have by causing their elbow to flare out while pushing the wrist downward, and then simply guide it across their chest, so you can slide out to their back.
This "lever drag" version can be helpful if your partner redrags (more on that below), or if it's difficult for you to break them down in the first place with the initial arm drag attempt. This time, make sure their elbow flares wide, but also be sure to shoot their wrist outside of the protective realm of their body. Next, fall to your side, bringing your partner with you, and setting up a different, lower path to the back. This can be extremely effective against a much stronger opponent against whom you might not otherwise consider an arm drag to be a viable attack.
Arm Drag Sweep
With the gi, you have a few different options for gripping. One such option is to go down the same path used for the closed guard arm drag series. This means starting with a same-side sleeve grip from your initial butterfly guard, and once you drag that across, you can sit up and take far-armpit control with your (free) left arm, hugging your partner in close. As you shoot your hips underneath your partner from this outside angle, you should find them to be very off balance to your left. Take them this way by using your hook and their momentum to finish a nice, controlled sweep.
The re-drag can be a great counter to the arm drag, but you're only likely to be able to do it a handful of times on your partner. Nevertheless, it's a great one to help understand the mechanics of the arm drag. When your partner pulls you in close, simply overshoot what they're trying to do by "helping a little too much." Once there, it's a simple matter of clearing their legs and switching your base toward their hips, ending in reverse kesa gatame.
As you play through these variations, find your own that work best, and—even better—try combining these approaches whenever you can. Use combinations of sweeps and threats to constantly keep your partner off balance, and be sure to remember to flow whenever possible. As always, let me know how these techniques are working for you!