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If you want a small horse for kids to ride, please don't buy a miniature. They're not bred or built to be riding animals. I would suggest a small pony, and one that is already broke. "Been there, done that" is a good description of what you should be looking for.
It's generally not a good idea for kids to grow up alongside young horses. They're unpredictable and dangerous, even as small ponies. It takes a LOT of experience to bring a horse along under saddle, and it's not something one should take on lightly. I think you and your kids would have a much better introduction into the horse world by taking lessons on a quiet horse from a qualified instructor that can point you in the direction of a good pony for your kids.
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