@TarsolyGer you can even go further than @DaveSch advised here, by completely eliminating the base path. If you do that, all relative paths start at the user’s home folder, and there is no OS dependent data stored inside the database anymore.
What I mean is, that when you use a path like
/home/enno/Documents/Archive
on Linux, you will also need a path like
C:\users\enno\Documents\Archive
on Windows.
If you include the Documents/Archive part in the individual media paths however, and empty the base path, Gramps will still find the media on Linux and Windows, because it automatically adds the user’s home path, when the base path is empty.
And maybe even that home path can be changed with an environment variable, in case you have your mother’s My Documents (with the internal name Documents) on drive D.
Any my guess is, that you can make this change on Linux by letting Gramps change all relative paths to absolute, then empty the base path, and change all paths to relative again. But that is something that I haven’t really tried myself, so you’ll have to experiment a bit, and use backups where needed.
The underlying assumption here is, that on both Linux and Windows, you actually have a Documents folder with that exact name. But even if you don’t, you can also use a folder with a different name, like gramps-media, as long as it sits in the user’s home folder.