Doing data migrations up front in a NoSQL system means you've changed the fields around in your document types, then you're done. Now 95% of your documents are wrong, missing those changes. Maybe you have linkages between different types of document (akin to a foreign key) - now a lot of those might be pointing to nothing.
In this model, it's the application's job to anticipate and work around these inconsistencies. Assertions that check for data integrity, if you have them, have to be modified to work around this. For a lot of the web applications we talk about these days, who cares - it's people's lists of friends and TODO notes. It's simple data and some dangling records aren't going to hurt anyone.
In the SQL world, we instead write migration scripts that migrate the structure and data all at once. This is a little more work up front, but as long as you stuck to a mostly normalized form and use great tools (which we now have) this is not a big deal ("MONTHS" to migrate across slaves sounds like they had some very wrong decisions made earlier on). The application can move straight to supporting the new structure and entirely forget that the old one existed. In this world, we can also have really complex relationships between fields, like when we're storing accounting or medical data linked to temporal records of changes. The application can consume this structure without worrying about consistency.