How do the Extended Cut flashbacks work?

Based on choices. Most of it is pretty obvious. However when it comes to order things vary. Whoever tells joker to go during the Crucible blast I think depends on who you brought with you most often and as for the slides it primarily depicts the big world-shaking outcomes and remaining ME2 squaddies that aren't directly related to any of the big world-shakers. When EDI/Shep/Hackett mention losses during the war you'll see the people that either died or were sacrificed and it varies on whether it's either of the 3 ending choices. You can find the official flowchart for Extended Cut somewhere. I believe they showed it at a panel launched soon after the release of EC.

The first one will always be Anderson.

The second flashback is someone who died in ME3.

The third one will be your LI. If no LI, it will be the one who Shepard has taken on the most missions as well as talked with throughout the game.

Here's a related thread

If you want to get technical, everything in Mass Effect 1, 2, and 3 runs off something called Kismet. Which is a visual scripting system for the Unreal Engine.

If a plot flag is set to true, then it will show a cutscene for example. If it's not, then it won't.

If Miranda is alive at the time that you make your final decision, her flashback will be shown. If she was killed at some point during ME2 or ME3, her flashback won't be shown.