Except for kernel modules - which are stored on disk and often loaded on demand. Though most distros keep around the old kernel and kernel modules to avoid issues. You can still get issues on distro such as Arch which replace the kernel and modules on an upgrade - though there is the kernel-modules-hook package with pacman hooks to keep the old modules around until the next reboot to fix this.

If you don’t have the kernel-modules-hook package installed on Arch you really should reboot shortly after an upgrade to the kernel or else new devices you plug in might not function until you do (typically USB as those tend to be what you hotplug the most).