Does Relativity of Simultaneity Imply Movement Before Action?

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Because of a the relativity of simultunaity, two people on a moving train both signing a paper will finish at the same time, but an observer watching the train go by will say that the person in the back finished first. Wouldnt this also mean that the back of the train is moving before the front of the train?
 
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Yes, but if the train is moving at a constant speed then that doesn't really matter.
 
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The rear of the train is further forward (closer to the front) than it otherwise would be. Doesn't properly account for the Lorentz Contraction, but interesting.
 
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