If you're always using Entity Framework migrations to make changes to the database structure, then you might be able to use the approach on the below. It depends a little on how your multiple databases are used in relation to the EF application.
Or you can use the EF migrations to generate a script that you can run on each server manually.
If you're interested in an automated process it might be worth taking a look at Deployment Manager from Red Gate (full disclose I work for Red Gate).
This lets you take a database and from Visual Studio or SSMS turn it into a package that you can deploy to multiple servers and environments. The starter edition for up to 5 projects and 5 servers is free to use, and it deploys to Azure. It can deploy .NET web applications too, but you can just use it for the DB.
It's very good for ensuring that the same database schema is on all servers within an environment, and for propagating database changes through test/staging/prod in line with any application changes. You can also integrate it with source control and CI systems to automate database changes from Dev to Production