I was looking for a way to automatically update the database after a deploy. The only way I found so far to do this, is to do this in code in your Startup class.
In the Configure method, use this code to upgrade your database other than Development:
if(!env.IsDevelopment()) { db.Database.Migrate(); }
If you make sure to set the correct environment variable in the environment you’re deploying to, it will upgrade the database with your migrations.