I don't think the OP is trying to play downloadable flash files (though members of BL can apparantly download these sound clips). The problem, I assume, is playing the clip on the webpage given in the OP - or similar pages on the BL site.

I'm now using Firefox on Android, and the sound clip doesn't play. I can see the waveform, but when I tap the play button nothing happens.

The reason turns out to be that this Firefox is using uBlock Origin as adblocker. If I disable uBO (on this site only) the sound plays. Firefox on my desktop uses AdBlock Plus, a different adblocker.

Flash isn't mentioned on the site I see in Firefox. I suspect you only see it on an old system, which does not support modern audio media.