From my experience, it seems that the host of a running live game can manually and instantly change it to PBEM.

However I don't know if/how it is possible to do so as I couldn't find any relevant information in the FAQ or help sections. Maybe I'm wrong and my previous games automatically turned to PBEM due to inactivity, as expected from https://www.thronemaster.net/?goto=community&sub=forum&fid=4&tid=2&pid=24654#p24654

Anyway, I would appreciate some help here

You mean how you turn the Live Game you host in PBEM?? Right next to "Reload Game" and "Review Game" in the above menu there's a small envelope which does the deal. You click it, get out, and Done!

It's also possible the other way as says the post... everyone leaves and after 15min it turns into PBEM

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Many people have died for their beliefs; it’s actually quite common. The real courage is in living and suffering for what you believe
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Silent Knight
World Cup - Team Argentina Captain

Right next to "Reload Game" and "Review Game" in the above menu there's a small envelope which does the deal. You click it, get out, and Done!
Whoa, I never noticed that before! That's exactly what I was looking for, thanks.