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Animate a point that follows a function inside a plot

I honestly couldn't follow your code, but I think I get the gist of your question. One thing that you need to take care of is what the range of the animated point is. You're just making assumptions ...
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How enforce a range constraint on an InputField for a Number as a Manipulate control

InputField does not have 'min/max' support but you can do: ...
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Animated contour plots inside multiple shapes

Define vertex failures: ...
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Two locators in separate plots using Manipulate

Try following: ...
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Animated contour plots inside multiple shapes

With your definitions, try the following, either with Animate or with Manipulate: ...
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