How can I inspect an element which disappears when my mouse moves away?
Hover over the element with your mouse and press F8 (this in Chrome) to pause the script execution. The hover state will remain in visible to you.
It take you to the sources tab. Go back to Elements tab. This time code will not disapper.
An alternative method in Chrome:
setTimeout(()=>{debugger;},5000);
Now you have 5 seconds to make your element appears. Once it appeared, wait until the debugger hits. As long as you don't resume, you can play with your element and it won't disappear.
Useful tip to avoid repeating those steps above every time:
add this as a bookmarklet:
javascript:(function(){setTimeout(()=>{debugger;},5000);})();
Next time you wish to use this, just click/tap this bookmark.
In Firebug there are different solutions for this:
Also you may want to follow issue 551, which asks for a way to temporarily block specific events.
Edit:
To find out which element it is you can also enable the HTML panel options Highlight Changes, Expand Changes and Scroll Changes Into View to make the element visible inside the HTML panel.
Sebastian