So it seems that you can (both in FireFox and Chrome) replace or change the properties of the image during dragover without affecting what is dragged, but that you cannot do that during dragstart. I suppose this is because the user dragstart handler is executed before the default handler. (Otherwise preventDefault could not work in the user handler.) Another work-around for this could be to set a timeout with 0 delay in the dragstart handler for removing the image, instead of relying on dragover to do that.
Another problem with the current implementation is that it uses the pieces as foreground image. As I mentioned, this caused the row expansion/contraction, at least on FireFox). In the Interactive Diagram we tested on this is probably not visible, because it already used an unnecessary large square size of 36x36 for the 33x33 images.
But there also is a second reason why this is problematic: in addition to highlighting the piece moves through a background color, the Diagram also has a mode where it uses marker symbols (dots, squares, triangles or stars) to indicate the various types of move. These symbols are then used as foreground image (i.e. cell content), displayed on top of the pieces (e.g. when indicating a possible capture). This mode is necessary to make the Diagram work on monochrome devices (like e-readers), where one cannot distinguish the colors. Below I posted a Diagram where I switched that on. (On monochrome devices it would get enabled by default.)
The problem is that it is the foreground image that will be dragged. But I suppose it must be possible to have the dragstart handler copy the backgroundImage URL of the clicked square, wrap it in an <img> tag and put it in the cell, and that this would still be in time for the browser default action to grab that image and start dragging it. The dragover (or timeout) handler could then remove the image again (possibly even put back the old cell content, if there was any).
theme=MV
firstRank=1
useMarkers=1
squareSize=33
promoChoice=CL
Pawn::::a2-h2,,a7,h7
Lance::C::a1,h1,,a8,h8
Cannon::mRcpR:Canon:c1,f1,,c8,f8
Lion:N::::1,1
King::::e1,,e8