It's also possible to create dedicate entries for each of the other KeePass vaults and set the URL field and password to the respective paths (i.e., "kdbx://PATH/TO/OTHER/DATABASE.kdbx") and passwords, then you can simply double click on the URL field to automatically open and unlock the other vaults.

The URL field in KeePass has lots of convenient features [0], but unfortunately they're quite "obscure" and not very discoverable.

[0]: https://keepass.info/help/base/autourl.html