I wasn't aware that Tumblr had an export function, and it looks like, officially, it doesn't[1]. There is a third-party migration tool to WordPress[2], but that's about all I can find. So unfortunately, not all is roses.
It would be nice if there were a standard format for storing blog posts and their metadata and attachments, so you could move from anything to anything. Perhaps the nascent Tent protocol[3] could do that when it's mature.
1. http://smarterware.org/8026/why-not-tumblr
2. http://tumblr2wordpress.benapps.net/
3. https://tent.io