Sword of Spirit
Legend
I mean, they've employed tactics like these before without a quibble and I just accepted it ("yes yes, your eldritch blast pushes someone into the moonbeam, they take damage"). Never before did anyone say "wait, he wasn't moved there!".
Lol! Yeah. The funny thing is that even DM's sometimes seem to put on those "player goggles" when they are in the role of player rather than DM. And this is with honest players, lol!
From memory, here's what the basic 2014 types of area of effect damage are and why they are that way.
Type 1: Area Discouragement. Moves Into + Ends Turn (no more than once per turn). This type is designed to give the opponents an option of whether they want to take damage or not. They can either go into or stay in an area and take damage, or get off my lawn and never take any damage at all. These can deal more damage than the other types because you are giving them a choice. The reason they don't just say "ends turn" and leave it at that, is because then someone could move into, attack the caster (or whatever) and then retreat from it on the same turn and take no damage, which would defeat the point. The booming blade cantrip is channeling the same intention. These are actually pretty interesting, because you can use them to get enemies to move where you want them a lot of the time. Think of fun ways to synergize that with your allies!
Type 2: Area Damage + Discouragement. Moves Into + Starts Turn (no more than once per turn). This type is designed so that opponents that can fit into the area at the same time will take damage once. After that it's their choice and functions about the same as Type 1 most of the time. You can also use other abilities to pull your allies out of this area, or push enemies into it, to prevent or cause damage, but it's not really a problem generally. Its balance is tolerant of that sort of smart play. This type is likely going to do less damage than Type 1 because you get the (almost) guaranteed initial damage triggering on their turn.
Type 3: Moving Area Damage + Discouragement Doom. Enters + Ends Turn (no more than once per turn). This is the big guns. It causes guaranteed damage in an area like Type 2, but the caster can move around and cause that damage to a lot more creatures than the area. After that, it basically functions like a turbo-charged Type 2 (despite having Type 1 phrasing). You get to choose whether to take any additional damage on your turn, but you can theoretically be pushed and pulled out of them again and again on different turns. These spells need to have pretty low damage (or be spells that don't deal any damage at all) to be balanced when you have a lot of enemies. They're the ones to really look out for and make sure that the design is intentional, and that you as a DM are comfortable with them. You might just want to house rule them into Type 2s instead.
Other Types: Probably Mistakes. Enters + Starts Turn (once per turn or not). This one would guarantee (short of pulling someone out) that you are going to take damage twice before you have a chance to do anything about it. That is never (or maybe almost never, I could be wrong) the intent. It's an even-more turbo-charged version of Type 3. Anything + Anything (NOT limited to once per turn). These are also probably mistakes. If they aren't mistakes. It's possible that something with really low damage is intended to work that way, but you probably want to look hard at it to make sure it seems intentional and you like how it works.
Wouldn't it be great if the DMG had told us this kind of stuff? I mean, we only learned about the fairly important design concept of "ribbons" that had used because they mentioned it in a UA. I don't know why they are so tight-lipped about the underlying math, design intent, etc. Is corporate afraid someone is going to learn all the behind the scenes design and make a better version of D&D than they did? Maybe they are! Maybe they might! Still ridiculous practice nowadays to treat the intent behind rules as a trade secret. Could also just be that the designers who know the stuff aren't particularly interested in spending their personal time explaining everything to us, and there wasn't space in the DMG. Next time someone catches a designer at a Con somewhere, ask them why they don't share more about the mechanical design process.)