Q: What muscles should I use for more control to smooth transitions from Down Dog to forward bend in Sun Salute A?
A: You are describing a very common phenomenon. The action that you are struggling with does involve the abdomen, but not the abdomen working in a vacuum. The important piece is to mentally connect the action of the leg stepping forward to an intention that includes pressing your hands firmly into the floor as a way of finding and emphasizing a lift in the lower belly.
My suggestion is not to go outside of the difficulty (no need to start working on crunches, for example), but rather to slow the action down and focus on the moment when it starts to get hard. One method that has helped me is to go through the following exercise: From Adho Mukha Svanasana (Downward-Facing Dog), rock your shoulders forward so that they stack over your wrists (as in Plank position). Then bring your right knee in toward your tucked chin and hold for five breaths. While you are in the pose, try to keep your shoulders over your wrists and very actively press down with your hands to create a lifting or scooping feeling in your lower belly. At the end of the five breaths, continue to press down with your hands and to relate this action to a lift in the belly. At the same time, as consciously as possible (we all have a tendency to check out when the going gets rough, when in fact that is just when our focused attention is most necessary), place your foot as far forward between your hands as you can. If it doesn't get all the way up, take another step or two until it does. Then step it back to Downward-Facing Dog and repeat on the left side.
Over time, you can integrate a version of this into the Sun Salute, no longer holding the knee in for five breaths but maintaining the intention and the relationship between the hands and the lower belly to help with the action.
See alsoFlow Safely: Protect Yourself Through Transitions