No the migration tool only converts from VPP based files to content based files. I would suggest you to use export/import to transfer content it will include blob as well. 

Even if the export UI currently does not support to explicitly point out folders or media but you can do it programatically though. you could either create your own instance of DataExporter or hook up event handler. Here is an example of the later http://world.episerver.com/Modules/Forum/Pages/Thread.aspx?id=71985&epslanguage=en

Is it just me or was the whole media feature a bit rushed...?

When we first upgraded some sites (mind you it was a supposedly stable release) the migration tool was just plain unfinished and didn't work. It was eventually fixed but the export UI still doesn't have a checkbox for media.

Matti, I agree that the migration tool is not release ready code, I have nearly never been able to use it without doing some changes to my solution to get the tool to work.
The last updates makes it better but, there should be a better solution for this.

Johan. Ok, to bad, for the first site this solution could be ok but you should build a tool for this that would be ready for when Azure/Amazon is public release.

I have not tried it myself but there is an open source tool to convert from file based blobs to any other provider:

http://world.episerver.com/Blogs/Per-Magne-Skuseth/Dates/2014/4/SQL-Blob-Provider-Store-your-blobs-in-the-database/

http://nuget.episerver.com/en/OtherPages/Package/?packageId=EPiCode.BlobConverter

I have found this nuget package, has anyone tried it?
https://nuget.episerver.com/en/OtherPages/Package/?packageId=EPiCode.blobconverter