I found that if I did a search in the comments archive page of the Moderation plugin (i.e. /wp-admin/moderation-admin.php?page=comment-archive), that the ‘Next Page’ link would take me back to the search page again, instead of to the next page of results.
It appears that the moderation_comment_archive() function is checking that $_POST is set and if not, it outputs the search form. Now, the “Next Page”, and “Prev Page” links are passing their args in the url, so I changed the occurrences of $_POST in the moderation_comment_archive() function to $_REQUEST (lines 1801 & 1856), and then I added &search=search to the Next and Prev page links, so they now start like this:
'<a href="moderation-admin.php?page=comment-archive&search=search&uid=' . ...
This seems to work. I haven’t checked the other archive pages to see if a similar fix is needed, but I wanted to post this solution in case anyone else is having similar trouble, and in case the authors might notice something terribly wrong with my fix.