How do I safely import MySQL data in a directory to a running MariaDB server?

Question

We're integrating another office into ours. That includes some of their data.

I have a new installed MariaDB 5.5 server. The data dir = /home/data/MariaDB. I've done a new/fresh install into that directory:

ls -1 /home/data/MariaDB
    ibdata1
    ib_logfile0
    ib_logfile1
    multi-master.info
    mysql/
    mysql_upgrade_info
    performance_schema/
    test/

The server is up & running.

I inherited some data from a now-defunct MySQL server. It's in:

ls -1 /home/data/MySQL-OLD
    auto.cnf
    ibdata1
    ib_logfile0
    ib_logfile1
    mysql/
    mysql_upgrade_info
    performance_schema/
    proj_DB_2136/
    proj_DB_4216/
    proj_DB_6217/
    test/

I want to move data for three DBs -- "proj_DB_2136/", "proj_DB_4216/", and "proj_DB_6217/" -- into my working MariaDB server. Same names, same data, everything.

I just need to do this one time, so I can start working on the old data on my new server.

I'd like to do it the right way, without losing anything!

mysqldump makes sense for me, but only for working databases already IN the running server.

What's the correct way to get this data safely brought over?