Thank you everyone. This is all helpful, and food for thought. I did this same project one year ago. It is a "memory making" day for families and children who have had a parent die at the hospital my wife works at, in a recent year. There were many activities for all, and I shot lots of other stuff (including horses running off by themselves with the notes tied to their manes, at the end), but the most popular and a very meaningful one for the children was the horse riding. I did some flash; none of the horses were spooked, but I certainly was not super close to them.
I have no control over what the kids and horses do, but there is no riding around independently. Professional horse therapy people are leading them around, one on each side of the horse. Still, I spent a lot of time running back and forth getting in position. Sometimes, the background on the less sunny side worked. Some of the blown out backgrounds were good; other times, what might have been a fine photo just didn't make it. Overall, I got a lot of amazing photos that were satisfying for everyone. (I wish I could share them here, to get more feedback, but considerations of confidentiality preclude that.) I was happy with the results overall, but I was well aware of the technical limitations, and I thought better flash might be a way to deal with it. Probably not, because among other reasons, even if I had infinite resources and assistants for setup, this is not a context in which the photography came first. I do now see, based on the very helpful comments above, that a faster lens is the likeliest solution; I will see what I can rent. I have two fast prime lenses: a Nikkor 50mm 1.8, and a macro, a Nikkor 85mm F3.5. I have occasionally used the 85 mm for portraits. I am probably showing my ignorance (but don't mind :-) ), but is it ridiculous to use a macro lens for this? The zoom is really so useful in this kind of situation, but perhaps I'll also try one of the primes on a second body (a D80). I haven't had good experience with ISO above 500, so am loath to go that way. I don't think I can get away with high ISO with a D7000.
How to hang all this stuff around me while I'm running around! Sigh. I think I figured out some way to hold everything to me with make-shift straps. Again, my thanks, friends, for your generosity with your time and insights.