Which lens are you using? I would think an f/2.8 or larger aperture desirable for such shots, and purchased a 135mm f/2 earlier this year, in anticipation of photographing indoor swim meets, after using a 100mm f/2.8 the last two years.* These are not inexpensive, but rental ("hiring" for those speaking the King's English) is an option. Raising the ISO is an option, but I try to keep ISO rather low whenever possible.
The goal, in raising the ISO or using a larger aperture, is to be able to use a fast-enough shutter speed to "freeze" the motion of the subjects. I would think 1/250th to 1/500th to be needed for horses, but having never photographed this subject matter, I may be wrong, as an even faster shutter speed might be needed.
The D7000 is fully capable of this type of photography, a most excellent camera! I use digital Canons, but my wife, an accomplished photographer, uses a D7000. I have considered purchasing a D7000, so that when traveling, we can economize logistics by being able to share batteries, chargers, lenses, flashguns, and such.
*The lenses I mentioned are Canons, and may not have Nikkor counterparts in the exact same focal lengths.
Canon 7D2/7D/5D/40D/1D2N; Nikon F6/D700/FM3A/1Dx/Coolpix A; Canon 40mm 2.8 STM, 135L, 50L, 35L, 50mm 1.8 I, 100mm 2.8L Macro, 10-22mm EF-S, 400mm 5.6L; Nikkor 50mm 1.2 AI-S, 50mm 1.4G, 50mm 1.8D, 16mm 2.8D Fisheye, 180mm 2.8D, 100-300mm 5.6 AI-S, 18mm 2.8D, 45mm 2.8 AI-P, Micro 60/2.8G; Voigtlander 90mm f/3.5 SL II