Because Maxwell's equations admit homogeneous solutions, they cannot determine a unique electromagnetic field from a given charge-current distribution until and unless suitable boundary conditions are imposed. And boundary conditions are, by definition, non-local.
To this I add, by the way, that Dirac and others have noted that a classical theory of point electrons -- even when augmented by suitable non-local boundary conditions -- is essentially untenable. A consistent classical theory of radiating charges has yet to be formulated.
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Why should I need to assume anything about the field at a distance greater than c t? (You seem more concerned about the mathematical structure of the theory than about the Physics the theory strives to describe. I am not)
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I recommend reading the Master: http://bit.ly/2yHQgmk
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