First up, apologies on my slow response. I've been having a little time off and am only back on board today...

FWIW, v14.x is based on Debian Jessie (TurnKey hasn't been Ubuntu based since circa 2011). TurnKey is binary compatible with Debian and uses unmodified Debian packages for the bulk of the OS. We just tweak some stuff (mostly security hardening, plus some other minor ease of use tweaks). Plus we also provide a few other "goodies" such as packaging Webmin (a web based admin UI tool), our custom backup tool TKLBAM and config tool Confconsole, etc. So essentially, TurnKey is a minimalist headless Debian install with a few less packages installed by default, a few tweaks to default config, plus a few extra packages provided by us.

As such, you can do an "in place" Debian upgrade on your v14.x servers to a Stretch base (i.e. Debian 9) by updating the sources.list.d/ files (from 'jessie' to 'stretch') and do your upgrade as you would on any other vanilla Debian (or Ubuntu) machine. Obviously all the same cautions and disclaimers apply to this process as per any major OS upgrade. So before you do anything, make sure that you have a tested and confirmed working "clone" or "snapshot" type backup of the server.

Although they're pretty dry and not a particularly exciting read, I recommend at least scanning through the Debian upgrade notes (covers Jessie -> Stretch). It's particularly worth being aware of the "Possible issues during upgrade". Also, whilst it's not directly related to the upgrade, it may also be of interest to read the "issues' section of the Stretch release notes.

Also, whilst Debian stable is now up to "10/Buster" (and you're currently on "8/Jessie") you should NOT try to jump from Jessie straight to Buster! You need to go from one release, to the next (i.e. not skipping any). And even then, I can't yet recommend that you upgrade a TurnKey server to a Buster base anyway. That's because we're running (WAY) behind schedule and don't yet have Buster packages available (although fingers crossed it won't be long). Stretch is still supported by the Debian security team though, so there's no real issue with holding off on a Buster upgrade.

Good luck and please don't hesitate to reach out for a hand if you hit any issues or have any troubles. I promise I'll reply quicker this time! :)