Next and Previous buttons do not work correctly
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Thunderbird 13  and later versions change the behavior of the Previous and Next buttons so that they work as expected regardless of whether or not the message is read. If you want the old behavior you would need to add the Previous Unread and Next Unread buttons instead. This article was kept to support anybody using older versions of Thunderbird.
There are four possible sets of buttons for moving between messages:
The Next and Previous buttons actually move the focus to the next or previous unread message. If you've already read that message you can't navigate there using these buttons and it looks like they're not working correctly.
You can workaround this by customizing the toolbar to remove those two buttons and replacing them with a "Next!" and a "Previous!" button which do what you expect. Download the Buttons! add-on and install it using Tools -> Add-ons -> Extensions -> Install. Right click on the toolbar near the top of the main window and drag and drop the Next and Previous buttons to the Customize Toolbar window, and the Next! and Previous! buttons from that window to the toolbar. Press Done.
The 'Back' and 'Forward' buttons act like a Browsers back and forward buttons. They are initially grayed out and are enabled as you've read enough messages to be able to navigate back and forth within that sequence of messages. If you find their presence confusing you can remove them by right clicking on the toolbar, selecting customize, drag and dropping those two buttons to the "Customize Toolbar" window and pressing the OK button.
If you want to keep the Previous and Next buttons you can change some of their behavior by modifying mailnews.nav_crosses_folders as follows :
See Modify Thunderbird settings if you don't know how to modify settings using the config editor.
Compatible by default
RealPrevNextButtons for new Thunderbird versions provides replacements for the Next and Prev buttons. It works with message preview, tabs and windows. Since it supports at least Thunderbird 5.0 it should work as is with any recent version of Thunderbird, due to the "compatible by default" feature added in Thunderbird 10.
Thunderbird 3 and later
RealPrevNextButtons provides replacements for the Next and Prev buttons that works with Thunderbird 3.*. They work with message preview, tabs and windows.
You need to install the add-on compatibility reporter add-on to disable version checking, in order to use either of these add-ons with recent versions. Ignore the warning message in the add-ons manager about it being incompatible, if you do that.