Adding to @JasonHill's suggestion on the Unread Posts Page, there are so many more pages that can be utilized in different ways listed here: https://community.lithium.com/t5/Other-Techie-Stuff/Lithium-Platform-URLs/ta-p/32509 (All credit to @AndyK for starting this list; bookmark it and please kudo that post if you like!)
I personally did not use the Unread Posts Page when I moderated, but if I did, I would go back and do a double check through each board for sanity purposes.
In terms of moderation process as a moderator, I would review our abuse reports and filter reports to catch any obvious violations before moving in board by board to read each post. When I had handled an abuse report or filter report, I'd lock it and post to it to let other team members know that I had addressed the report.
Additionally, we made usage of the Moderator Flag column (Board Options > "Show Moderator Flag Column") to track which posts had been read by moderators.
One last suggestion is that you can float an individual topic just for your viewing (Board Options > "Float this Topic for Current User") and that will only float/pin the topic to the top of the board for your viewing. Noting that in case you may find that function useful for posts you want to go back to on a specific board.
There are many options to moderate in the platform, and I know I've utilized the great suggestions from @JasonHill, @DeannaB, and @SandyD when working as a moderator on Lithium before.
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