mysql – Moving a large MariaDB database between MariaDB servers

Once my DB size is multiple hundred GB in size, it is not practical to use the documented methods for transferring a database to an alternate db server due to sheer amount of time (days to weeks) for that slow process. The database is primarily located within a directory tree structure which can easily be copied, however, there seem to be files at the “data” directory (ibdata1, …) not unique to a specific database, that contain some information that is required to successfully xfer to a new location. Is there a way to extract the needed information from this file or set of files and provide to the new dB server? Or is there some way to cause MariaDB to maintain each database within the appropriate directory tree, thereby permitting file transfers to function?
Some background on my objective:
I capture data to my databases on a specific server, but that server is limited to only 1TB disk size.
Once the space on that disk reaches a threshold level (say 10% remaining), I want to move that database to an alternate server with much greater storage available, and free the space on the capturing server.

Stating the objective a little differently: I wish to take a db from one server(1) and add that db to a different server(2), allowing all tables in the “copied” db(1) to be emptied, freeing space for additional collection. (but do so in finite time — minuites/hours instead of days/weeks)