This has been the case for a long time, and my workaround has always been to click on a portion of the item near the right-hand side, still inside the item rectangle but not over the link text itself.
This can be a very small area depending on the length and wrapping of the link text.
This is related to a new bug that just came up recently, in which even items that are just plain text also provoke a pop-up menu about sharing or downloading (sharing or downloading the whole page I presume).
@charliehamu Talking about the new bug - could you please try how it works now? m.checkvist.com should be updated. After that, I hope, drag and drop would work as previously.