There is no widely common procedure or guideline for moving the actual comments from the main site into a Meta discussion. It varie a lot between sites, some sites preserve a lot of comment streams while others only preserve some of the particularly noteworthy ones. Some put it in chat and link the transcript, others post into a post body.
There is no actual "move" functionality. In terms of simplicity, the best option is traditionally a whole-sale copy/paste of the comment thread into an appropriate medium, then the moderator "Delete Comments" ability to clear the thread. You'll need a moderator to accomplish the latter, but a normal user could always do the first part and then flag for the second.
As far as when to do it, it depends a lot on the nature of the comment exchange. Generally, Meta discussion (or a move to chat) is sprung when things are no longer suited to sticking in comments. This often comes up because:
- The comments are getting too heated
- The comments are straying away from the content of the post they are attached to
- The comments are numerous in quantity
- It is a reasonable expectation that one of the above is going to happen if nothing is done
In general, preserving the comments in the Meta discussion is only necessary if you're getting rid of the comments on the Main site, and only if the comments are worth preserving. For example, if it's a heated discussion, then keeping all the flames alight is probably not wise. Or, if you're just pre-empting a thick discussion, then adding a comment "This has now had a Meta discussion opened, please direct further commentary there" can let you simply leave the 10 or so comments on the post, if they're fine on their own right.