I discussed this with Risker. There doesn't seem to be anything coming up in the JS console. She observed that something is being forbidden by Kaspersky around the time of the dots showing up. I'm wondering if the images we load into the popups might be triggering that. Hard to know if it's the source of this problem though.
(copied from the enwiki feedback page, so you'll have it here)
I did some work tonight with Krenair (thanks for taking the time, Krenair!) and we worked out that there were a few things happening. First, it appears that my anti-virus software (Kaspersky) was blocking the popups that should come from the blue dots. I did a bunch of tweaking with that and with the browser, and eventually found a combination that allowed the popups. I don't think I'll leave things at these settings, though; they're less protective than the "factory settings" and in just the few hours since I did that I've had quite a bit more spam and other junk showing up. I also haven't had any problems with other popups on Wikipedia, which were able to show up through the AV software, so it is odd that these particular ones got caught. But, and this is an important but...even with all that, clicking on the blue dots did nothing for me. However, clicking on the actual link symbol or the word "cite" resulted in a popup. The popups had a brief bit of text describing that one can make links to internal or external web pages, or that citations improve your content. Neither described how to use the button, though, and I needed to click "Okay, got it" to dismiss the popup. I could, however, edit with the popups in place, just as long as I didn't want to edit anything that was covered by them. These were also pretty big popups and most of the space seemed to be taken up by images that weren't great illustrations, but I realise this is an early iteration. (I won't even go into my usual tirade about live content production being not the right place for alpha testing.) But this needs a lot of work. And the flashing dots still don't do anything, the popup is actually on the button.
Thanks again for taking the time at what was probably well past your working hours, Krenair.