Unfortunately, content is not backwards compatible (out of the box -- you could potentially use a third party tool to migrate but you likely won't achieve the results you are looking for without a lot of work).
You will first need to upgrade the second farm (which is running the June 2010 CU) to the same or a higher version (October 2010 CU or anything else newer).
Once the versions are satisfied, you can perform either a database attach or site collection backup and restore. Assuming the farms are in the same domain, have the same solutions deployed, and DNS/URLs are updated to reflect the move, it should be pretty
seamless for your users as both of these methods copy everything.
