By Digi Cam, I guess you mean compact camera or bridge camera. Also what do you mean by not crazy money - an idea on budget would help.
Essentially you'll need some for of manual control, be that Aperture / Shutter Priority or full manual control.
I'd say horse head shot, in focus at 60 - 90 feet you'd need a 300mm lens, shooting at a minimum of 400th sec. You'll need to illiminate camera shake too.
If you're trying it on a compact, shutter lag and AF will be you're biggest problems., in my view.
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