It depends on what you are using to host/run your database, for example MySQL has a tool called mysql dump that let's you effectively backup or pull the live data to a local machine. If you are you using something else like PostgreSQL, then I am sure they have a similar tool, if neither of those are applicable you could provide some more information but what you're using. As far as if there is an easy way with Railway specifically, I know they do have a production/development environments that you can create, I have not played around with that too much but I am planning to soon. When you go into a project at the top it should show production and you can click it and create a new environment by copying your current environment, that might be what you are looking! (However it doesn't look like it copies the data from what I can see, so you might have to do a combo, i.e. setup the staging environment, do mysqldump (or the same tool for whatever database you have) then remote into your database services and restore the database via the dump file) Hope this helps if not again just provide more information.

Yes, I have staging + prod environments set up and just hoping for a simple railway-specific tool for migrating database volume data between environments without having to pg_dump. But I'm guessing this isn't a thing!

Yea from what I can see as well there is no "one-click" feature if you will to do that. You have to dump the current database, then restore that into your other environment, though it shouldn't be more than a couple commands since Railway provides you with a direct way to connect to the database services. That would be a very cool feature too see them add, though would most likely be a challenge since they will most likely have to have different implementations for each services, i.e. postgres, mysql, etc...

The Following is an option or way which is not one-click but much closer to one-click.
--> pgAdmin / TablePlus / DBeaver (GUI tools)

First Connect to both Production and Staging using Railway’s database URLs.

Second, Right-click → Export/Import or Copy Database.

Much simpler than running pg_dump / psql manually.
The ONLY CAUTION is It will overwrite staging data unless you selectively copy tables.