is it known to play well with Kotlin Multiplatform libraries?

TornadoFX isn't a GUI library itself; it's more a Kotlin wrapper around JavaFX and a small RCP framework. As such, it's not going to run in Kotlin Multiplatform- it has a hard dependency on JavaFX. To do web you'd probably need to look into https://www.jpro.one/, and for mobile https://gluonhq.com/products/mobile/javafxports/ both of which are going to carry their own quirks you'd probably prefer to understand when getting started rather than trying to adapt to later. At that point I don't know if Kotlin Multiplatform is going to help you in a meaningful way, over just using JVM Kotlin. Maybe you could also look to building a native GraalVM image? My goal is to support desktop and Android; and I don't have experience with jpro, gluon, graalvm. I tried Jpro once quickly and can say it didn't "just work" and at the moment I'm just focused on desktop anyway. My plan from the start has been to use Clean Architecture and anticipate that tornadofx is just going to be for the desktop UI, in which I'll reimplement the Android UI in native Android when I get to that. Without current needs of an iOS port, I've been pretty liberal in my use of Java dependencies.

Well the idea I had was that the Tornado/JavaFX code would reside in the jvm side but call into Kotlin multiplatform libraries in the same way iOS and Android would.