Yes, the Kindle does this and it is one feature I really miss on Kobo. My Kindle has an email address (e.g.
me@free.kindle.com, free meaning use wifi, which is free delivery, as opposed to using a kindle's 3g connection which may cost you money for the transfer). If I send email to my Kindle email address and attach a .mobi file to that email, it will magically appear on my Kindle the next time the Kindle is attached to WiFi. This is also the method used to save web pages to the Kindle using Instapaper, for example.
This is a great feature, and Kobo needs it.