Actually, Mike, I consider each of the throws you listed to be representative of very different "sub-families" of ashi-waza.
As you've probably figured out, I like putting things in groups. I think it makes it easier to learn, teach and understand if things are related to each other.
So I divide sweeps into two families- pinned sweeps and floating sweeps. Floating sweeps divide into sub families of extension and cross sweeps.
Pinned sweeps (eg O-soto and O-uchi gari) you commit uke's weight to one foot and then remove that foot. If balance is properly broken on O-uchi weight can be committed to both. The most common screw-up for O-uchi is trying to sweep a foot that is in the air or has no weight on it.
Floating sweeps involve moving the foot out of the way in the instant before weight is committed to it. After the center of gravity has shifted toward that side but before it contacts the ground.
There are only two good directions to do this (plus one third option, the stop). You must either sweep the foot across the centerline (de-ashi), or extend it into the front splits (ko-uchi). If you try to extend it to the side splits, you just make the base wider and are trying to work against stronger muscles.
The stop is like sasae tsurikomi ashi- the foot is frozen at the centerline and the CoG is pulled over it.
As for the hands- In o-uchi, either hands or timing can set up the throw but hands are critical to pin the weight down on the foot you are sweeping. Follow through is best with a fully committed shopulder and hip.
In cross sweeps, working hands and feet together for 2-way action is critical on the follow through. Also, you can buy yourself a little more time (and timing is hard on all floating sweeps) by pulling forward slightly on the sleeve/shoulder of the advancing foot.
Lastly, in extension sweeps, a subtle forward pull can really extend your window of opportunity. Done gently enough, uke just feels a need to take an extra long step and his foot never quite reaches the ground. It also helps to be able to fake the feel of a backward step with your shoulder.
Just my little concepts. Is this what you wanted, Mike?