In Firefox and IE how can change the cursor while dragging over different targets?

I'd suggest using jQuery ui -- droppable and draggable.

You can hide the default/actual cursor using cursor: none and use a stylized DOM element to render your own 'cursor' element.

I've provided an example here: http://jsfiddle.net/lawrencealan/WMyu7/78/ (updated 6/8/2017)

Note: mousemove stops firing during dragging of "draggable" attributed elements.

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EDIT: As of September 2019, the example is now broken.