Good news first, it's normally totally possible without any issues.
AFAIK your info about specific modules and kernel drivers is wrong. Except specific updates, macOS includes all drivers for all supported models.
Licensing info on the other hand has probably to be re-entered.
What I do in cases like yours (backup your data first, just to be safe):
- Create a bootable install medium from the latest OS you want to install (full installer from Mac App Store)
- Clone/Move the SSD to your new computer (don't yet boot from it)
- Boot your new device from the created installer and install the OS OVER your existing system
This way, all your userdata, settings and apps remain intact, while also ensuring complete and clean system files.
If your Mac Pro isn't on the latest firmware yet, first do a clean install and apply all updates (enough guides here). Afterwards proceed with my suggested steps.
If you want to stay on High Sierra, 10.13.6 is the latest full installer.
For Mojave you need a Metal-compatible Graphics card.
For Big Sur and Monterey your furthermore need Open Core.