Disagreement over agreement

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A member of an editors' group on Facebook posed this question yesterday:

"Peristalsis is a series of muscle contractions that move/moves along the digestive tract."

Move or moves?

There followed a something less than edifying exchange of views and comments, some picking move and some picking moves as if we were in a schoolyard choosing up sides for Red Rover. 
One gentleman insisted repeatedly, and dogmatically, that it has to be moves to agree with series. He knows how to diagram sentences, he said, and his judgment is "based on more than three decades of experience that started in the newsroom of a metro daily."
Well, I, too, can diagram sentences and have had more than three decades of experience at metro dailies, so we can throw the Argument from Authority over the side. 
I, at least, presented an explanation for my choice: "Series is the predicate complement of peristalsis. Peristalsis is series is the main clause. Of contractions is a phrase modifying series. That move is a subordinate clause modifying contractions. That is the subject of the verb move, and the question at issue has to be what the antecedent of that is. Contractions is the nearest antecedent.* 
Where the gentleman with the mistaken belief goes wrong, I think, is to confuse the syntax in this example with the common error of mistaking subject-verb agreement when there is an intervening prepositional phrase.
Example: The fate of the banks remains in doubt. Some writers, seeing banks adjacent to remains, think that the verb should be remain. But fate is the subject, and the prepositional phrase has no bearing on agreement. 
But the sentence we started talking about is a different matter, an independent clause followed by a relative clause. One has to determine the antecedent of the pronoun introducing a relative clause to determine whether its verb should be singular or plural.
The one/one of those examples will show how that works.
If you write, She is one who washes clothes in the creek at the foot of the hill, the antecedent of who is one, and the verb is accordingly singular.
But if you write, She is one of the women who wash clothes in the creek at the foot of the hill, the antecedent of who is women, and the verb must be plural. 
English usage can be full of tangles and thickets, and experienced grammarians regularly disagree over subtleties and intricacies. The introductory example, however, is neither intricate nor subtle. 
*God help me if Geoffrey Pullum or some other grammatical heavyweight shows that I've got the wrong end of the stick. I'll have to eat crow publicly, and it is a dish that, though often served up, has never been to my taste.