Swapping bikes with different drivetrains on Kickr core.

An 11 speed chain works fine on a 10 speed cassette (the issue would be that your shifters are indexed for 11 speed, not the chain width), and a 9 speed chain would also work reasonably on a 10s cassette.

9 speed chain however would not work especially well on the 11 speed cassette because the outside width of the chain will be too wide, it will be quite noisy. An 11 speed chain will work OK on a 9 speed cassette notwithstanding the indexing issues (the inner dimensions of bicycle chains are all the same - it is the outside dimensions that have changed, i.e., the roller pins are shorter - the velonews article linked below gives more info). Cassettes are usually OK with a narrower chain but less so with a wider one - 1s wider usually works but the wider you go the more issues you have.

A 10s chain would work fine on a 9s setup in general (in fact many people report it runs quieter like this), and your 11s chain will work fine on a 10s cassette, so what will probably work best is:

10s cassette on the Kickr Core
10s chain on the 9s bike (all the time)
Keep your 11s as is

But it won't work perfectly whatever you do due to the indexing of the gears - there will be some ghost shifting and so on that you won't be able to get rid of. My turbo bike is horribly set up and indexed though and it hasn't bothered me that much, so you might be able to put up with it - especially if you find a single gear that works and you are using erg mode

https://www.velonews.com/gear/tech-faq-chain-width-explained-compatibility-queries-answered/