Twenty of the 200 aid trucks from Russia have crossed the Ukraine border, Reuters witnesses say.
The trucks are said to be being accompanied by a small number of pro-Russian rebels.
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A 55-year-old man has died after being shot in front of shoppers in a packed supermarket car park in broad daylight.
The suspect, 35-year-old Willie Corey Godbolt, has been arrested and was being treated for a gunshot wound in hospital.