If anyone else got this message:

We don't do this often, so you know it's important: your Fluxer client is getting old. Like, "soon to be end-of-life" old.

It's because the Fluxer Flatpak from Flathub is basically abandoned... for now anyway. No one is updating it because all the dev work is focused on canary.

However, there is a Fluxer Canary Flatpak (though not from Flathub): https://github.com/yofukashino/app.fluxer.FluxerCanary

flatpak remote-add --user --if-not-exists fluxer-canary https://yofukashino.github.io/app.fluxer.FluxerCanary/index.flatpakrepo  
flatpak install --user app.fluxer.FluxerCanary