Does KiCad run on Linux?

KiCad's download page offers Windows, macOS, Linux and a container image. On the Linux side there are AppImage downloads, which are single files you mark as runnable and then open, and packages for Ubuntu, Fedora, Debian and other distributions. The Flatpak version uses the id org.kicad.KiCad and is listed on Flathub, so it turns up in your app store and updates itself.

Linux is arguably KiCad's home ground rather than an afterthought. The same version numbers ship on all three systems at the same time and the program looks and behaves identically.

Your projects come across as they are. Schematics, board layouts, symbol libraries and footprint libraries are the same files on every system, so copying the project folder is the whole job. The one thing to tidy is any library path that points at a Windows folder, since KiCad will ask where those libraries went the first time you open the project.