Hey David,
Ideally, it's best that you create an export dump through phpmyadmin and export the tables over. Here's the general guideline that you could use and it is also what we used previously when we migrated from current Joomla version to Joomla 3.4.
*Backup your data first before perform migration
*Migrate all the current Joomla version user to Joomla 3.x first using 3rd party extension (make sure all the user id is same as current Joomla version)
*Don't Install Easyblog in your new site.
1. Open up PHPMyAdmin tool that is provided by your hosting provider and open up your current site's database.
2. Make a sql dump for all the tables that has the prefix of #__easyblog_*
3. Download the .sql file and save it on your desktop.
4. If your new site's database prefix also uses the same prefix , then you can skip this part. However, if you are using a different database prefix, you'll need to edit the .sql file that you've just downloaded, and do a search and replace of jos_ with your new database prefix.
5. Open your PHPMyAdmin and this time, open up your new site's database.
6. You should see an import link at the top. Click on that, and click on browse. Select the .sql file and click the run import button.
Note : 7. Install the latest EasyBlog on your new site Joomla 3.x
**migrate additional - if your blog post have including the image file, you have to do this step as well.
That was to copy over the image and media files associated with EasyBlog (e.g. the "/images/easyblog_avatar", "/images/easyblog_images", "/images/easyblog_shared", and the "/media/com_easyblog" folders and all of their contents).
Hope this help.