I'm running Kubuntu 18.04 with Plasma 5.12.7. I've configured the Compose key and bound it to Right Alt. Now I can type extended characters in Kate for instance or in LibreOffice and even in konsole, but not in browsers. I think since this is a wiki page, it should cover this major case.
I managed to get it working in both Firefox and Chrome by providing QT_IM_MODULE=xim prefix, but I couldn't make it permatent since I haven't found /etc/X11/xinit/xinput.d/default file, referenced in the page (basically, I don't have etc/X11/xinit/xinput.d/ directory on my system - Kubuntu 18.04).
Also, I wonder is there some other way to just get it working in all programs without this mumbo-jumbo with editing system files? Say, having something in the UI to tick/untick would be a pain yet, but a lesser one.
I don't think this is a Plasma issue. I use the Compose key a lot and have never seen this issue either in Firefox or in Falkon. I use Neon, so perhaps it is a Kubuntu issue; or perhaps you have assigned the compose function to a key, that browsers treat specially? Do you have the Plasma Integration addon installed in Firefox?
If you find out what the problem is, we welcome feedback so that we can update this page.
PS. Have you sees the other thread on this Talk page? It seems to be related, and a solution was found: scroll all the way down - you may have to click a link saying something like "Show 4 answers" to see the last entries.