DB schema migration script frameworks (at least in Python, Ruby & Java lands) do typically support both upgrade and downgrade directions. People skip implementing and testing the downgrade side if the development model doesn't need it but the problem of what happens to the data is controlled by what you put in the "down" migration script.
I'd guess if you can't throw the data away, you won't do a down migration, you'll do an up migration that changes the db to save that data in your preferred way before undoing or reworking the previous schema change.
I'd guess if you can't throw the data away, you won't do a down migration, you'll do an up migration that changes the db to save that data in your preferred way before undoing or reworking the previous schema change.