Why do some of the blocks have the "Move Blocks" icon, and some do not?

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Because these blocks were created in the different ways. If the block has a "Move Blocks" icon, it means that the block was created by the user, not by the automatical service and it can be dragged. If the block does not have the "Move Blocks" icon, it means that the block was created by the service and can not be dragged.