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 old site to new site.

*Backup your data first before perform migration
*Don't Install Easydiscuss 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 #__discuss_*
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 your old site current Easydiscuss version on your new site

Hope this will help.

What if the new site already has the new forum installed and the users have been migrated already

Make sure your new site forum do not have any important post, then you have to uninstall Easydiscuss extension from backend > extension > manage > search `easydiscuss` keyword > click uninstall

Then open your new site database > select those #__discuss_ table > drop these table.

After that follow my above step start to migrate over your new site.

Note : Remember do the full backup on your old and new database.

Will doing these steps damage the users that I have already moved over to the new site?

I believe I have misunderstand here.

Hello, I am hoping to move my forum to a new site I built. I have migrated all the users and installed the forum on the new site, I just need the posts moved. Is that possible?

From your first question, you mean you already migrated all the user into your new site is it under your Joomla user which stored in this table #__users?
If yes, that is not affected your user data because you only drop these Easydiscuss table (#__discuss_xx > refer my attached screenshot below) instead of the rest of the default Joomla table.

Or do you mean you already migrated this Easydiscuss user table #__discuss_users into your new site?
If yes, it will not affected with your user data because you should migrate all of your old site existing Joomla user data over your new site.

By the way, may i know how to you migrate your Joomla user data into your new site?