I have experience with a Fujitsu Lifebook U9310x.
My general advice would be, if you’re going to use Linux on a convertible, install Fedora. It has the best and newest implementation of Gnome, and Gnome has the best support for convertible, touchscreen and on-screen keyboard support.
On Fedora, the experience was almost as good as on Windows, whereas I had issues with Debian not correctly switching modes when I fold the keyboard back, not popping up the onscreen keyboard and not correctly rotating the screen.
Disclaimer: I haven’t tested Ubuntu because I personally dislike it. But if it’s certified for your hardware, that would be the first thing I’d try.
Hardware advice: Don’t get a Fujitsu Lifebook U for writing. The keyboard sucks badly, to the point where about every 50th keystroke simply doesn’t register. There’s a Lifebook E convertible now which is more budget-friendly and has a better keyboard, but it’s too thick and heavy for use as a tablet.
Thinkpads consistently have the best Linux support, so that’s what I’d have bought if I hadn’t got a 60% discount on the Fujitsu from work.