I'm trying to move my database from one server to another, basically to allow the Player Ranks plugin to pick up playtimes and assign ranks automatically.

I am not sure how to transfer it and ensure that the player ranks plugin will pick up the existing times.

If this is in the wrong place let me know and I'll post in the right place.

Thanks in advance

Copy the plugin's files from oxide/data to oxide/data on the new server while the server is shutdown.

Once I've done this, the player ranks mod should pickup the new playtimes automatically shouldn't it?

Generally, yes. Same concept for all plugins.