Blocks should be draggable & rearrangable without having to click into it. But the UI has been noted to feel heavy. Let's try a lighter UI in a separate version of the UI prototype:

In this UI, mousing over the block shows the left border, as well as rearrangement controls.

The border on the left also indicates a draggable area, that invisibly extends a fair bit to the left of the border. This is only surfaced when you mouse over, as it's a desktop enhancement. So the invisible area should have a grabby hand cursor:

    cursor: move; /* fallback if grab cursor is unsupported */
    cursor: grab;
    cursor: -moz-grab;
    cursor: -webkit-grab;