I teach high school, so I don't really get a say in what kids I get. Scheduling is so complicated and there are so many moving parts that I just get who I get. I would love a draft, would be funny and we have joked about it before.
I do know some elementary teachers and have heard different things at different schools. Some, it's just random and there is no changing (no matter how much a parent or teacher begs). Some schools randomly assign and then if teachers want to trade, administration is fine with it. Some schools do it random and then if you are a parent or teacher that is either well liked/respected by administration or a big enough pain in the ###, kids can be moved around. I have never heard of a draft. Sorry. If your kids keep ending up with the same teacher, here are the possible reasons why:
1. Those teachers really like your kids and request to have more of them
2. Those teachers are hated by administration and your kids are awful so it's a punishment
3. Those teachers are hated by administration and you are really annoying parents so it's a punishment
4. The administrator believes in the family-teacher relationship and likes to keep siblings with the same teachers
4. Just randomness