All members of the board should have the ability to edit and move cards (unless they're observers, which is a business class feature). If your team members aren't able, then it's likely that they've been added only to the team as members, and not added to the board itself. Being a member of the team doesn't automatically add them to the board, so they'd need to go onto the board (from the team page) and click the join button to the right side of the board in the menu. Hopefully this is the situation and will be easily rectified.
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