Use any service you want, Syncthing, Dropbox, etc. Keepass is smart about saving. If the DB is modified in the background while it's open, it will merge your changes rather than overwrite.

Personally, I keep my file on Google Drive and download it wherever it's needed. It does require a bit of manual tracking to ensure I've got the latest file, but I've only got about a handful of devices that I need it for, and even if I don't have the latest version of the file, it just means I might be missing a password for a particular service, and I can quickly download the latest anyway.

Obviously it's not as nice as having a cloud service, but it's open-source and doesn't require trusting a third party, which I like.

it doesn't, it's up to you to manage that. you could use rsync, dropbox, google drive, icloud drive, or whatever mechanism for syncing you like.