Pete

Note: Obsolete since WordPress 4.4
<a href="https://core.trac.wordpress.org/ticket/32230" rel="nofollow ugc">This feature has been added into the core of WordPress 4.4</a>. You can now export your attachments by going to Tools->Export and select ‘Media’ as the export type.
Original Description
WP Attachment Export allows you to export your media library into a WordPress eXtended RSS or WXR file. You can then use the Tools->Import function in another WordPress installation to import the media library.
<h4>Issues / Bug reporting</h4>
Found a bug? Please use the issue tracker at <a href="https://github.com/thehelvetian/wp-attachment-export/issues" rel="nofollow ugc">https://github.com/thehelvetian/wp-attachment-export/issues</a>
<h3>Usage</h3>
User <a href="https://wordpress.org/support/profile/philho" rel="ugc">PhiLho</a> has put together a good summary on how to use the plugin at <a href="https://wordpress.org/support/topic/good-but-could-use-more-explanations" rel="ugc">https://wordpress.org/support/topic/good-but-could-use-more-explanations</a>. Thanks PhiLho!

WP Attachment Export

works perfectly.

Exports only posts of type 'attachment', i.e. your media library

ryanpcmcquen

Makes local copies of all the linked images and pdfs in a post, adding them as gallery attachments.
Source & support:
https://github.com/ryanpcmcquen/import-external-attachments
<h4>Credits</h4>
This plugin is based on the work done in the “Import External Images” plugin by MartyThornley.
https://github.com/MartyThornley
HTTPS support added by IvanDoomer: 
https://github.com/IvanDoomer
PDF support added by bengreeley: 
https://github.com/bengreeley
Most of the JavaScript was rewritten from the original plugin, to reduce the 
number of global variables.

Import external attachments

Makes local copies of all the linked images and pdfs in a post, adding them as gallery attachments.

WP Attachment Export vs Import external attachments