I created an Outlook Addin that allows you to drag and drop from Outlook to a web browser using the method outlined in wimix's answer here:
Drag & Drop directly from Outlook in web app
Basically I hook the Ole32 DoDragDrop function, save the data to file, convert the data to the CF_HDROP format and then call the DoDragDrop function again but pass the a new IDataObject.
Everything works as expected with IE and Edge. The only issue I'm having is that when dragging to certain pages in Chrome it does a "move" and deletes the original email. It seems to be related to the HTML5 DataTransfer.dropEffect and DataTransfer.effectAllowed properties. If they are not set or set to 'move' then that feedback is sent through to Outlook which deletes the original email.
It almost seems like that is how it should behave but I've tested this on other addins that allow drag and drop to browsers from Outlook and they don't seem to have the same issue. So I feel I should be able to fix/control this behavior in the addin but I'm not really sure how?