I am using EF 6.1 and I use code first approach with an existing database with data in a production environment.
Is it possible at all to migrate model changes and keep
Yes, however, it can be complicated depending on specific case (on complexity of changes):
Overview: https://docs.microsoft.com/en-us/ef/ef6/modeling/code-first/migrations/existing-database
Read this: https://docs.microsoft.com/en-us/ef/ef6/modeling/code-first/migrations/index
And then this: https://docs.microsoft.com/en-us/ef/ef6/modeling/code-first/migrations/teams
Update: updated the links based on redirects.