The V6 Auto will use the latest version of the Lotus IPS (Intelligent Precision Shift) setup, with Toyota (Aisin) torque convertor auto box and Lotus controller software.
In normal driving in full auto mode, the system decides when to change up and down based on speed, engine load, fuel economy etc. It works very well, the shifts are quick (not as fast as a DCT but still faster than a manual) and are done basically where you would probably change if driving a manual. It'll change down if you kick down the throttle for an overtake and it'll handle stop/start traffic easily.
At any point, if you use the paddles to override the auto then it will change up/down per your selection but will stop you doing stupid things such as dropping it into 1st at 100mph or 6th at 10mph. It'll also take over if you're braking and not changing down.
In Tour driving mode, if you use the paddles then the system will revert to full auto mode again after (I think) 30 seconds. So you can do 1-2 changes and then let it take over again.
In Sport driving mode it will stay in manual mode once you start using the paddles. It will auto-blip on downshift for you, which makes you sound like a driving god. In auto + Sport it will hold the gear to higher revs, change quicker/more aggressively and do all the auto-blip drama on down-shifting.
I've driven the same setup on the Evora 400 and recent GT410 and (don't tell the manual aficionados) it's good fun! If you can find an Evora IPS to test drive, you'll get a very good idea of how it works on the Emira.
There's more detail on the controls in the Emira handbook, which is on the Emira section of the Lotus website.