I don't think it will automatically kick you out of the tournament if your rating goes outside of the limits before the tournament starts. The tournament director can kick you out of the tournament though. Some TDs are vigilant about kicking people out. I personally don't think a few points over or under the limits I set for my tournaments is a big deal.
Tournament Rating Limits
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If a tournament lists 1400 and below and you are under 1400 when you sign up but rise above 1400 before it starts, does that disqualify you? Or does your sign-up rating still stand?