Move to Start by Philipp Kewisch

Firefox already has a move tab to start feature. Using that however retains focus on the tab you moved. This add-on will move the active tab to the start, but change focus to the tab originally next to it. This way you can triage your tabs one by one.

By default, the add-on will also unload the tab that is being moved to start. You can disable this in the preferences.

You can also use the keyboard shortcut Ctrl+Shift+Left Arrow as well. This may not work for all tabs (notably Google Docs), you can change the shortcut by clicking the cog icon in Tools > Add-ons.