What does “Advance a Move” mean in Urban Shadows 2nd edition

I can't imagine running any campaign that isn't improvisational to some degree. I am running a Dungeon World campaign which is more traditional in campaign design because my players come from more of a D&D background, so I'm working from that perspective. That works fine- effectively my fronts are running in a sandbox and the players get to choose where they go and what they do but they often look to me to give them a direction when they aren't sure where to go next or what to do. My goal is not to restrict the agency of the players or the power of the moves, however - if anything I'm trying to guide them into more of a DW mindset - but the consequence of the set-up is that the game we are playing is carried more on my shoulders as a GM in narrative and worldbuilding terms rather than the constant collaborative efforts that characterise the game in its purest form.

I think the biggest element to this is the fundamental guideline of "Draw maps, leave blanks" - I apply that to everything I am doing as the game develops. I know in general what might happen, I know what the various fronts and groups in the world are doing in general and then we learn the details as the PCs interact with them. Then I can set up problems for the players to solve without knowing how they are going to actually solve them- that is where the improvisation really kicks in.

To look at the things you are interested in:

• Elaborate setpieces - these are often "grand finale" type situations, in which case if the players get towards the location of that finale you have an opportunity to bring them into play. In general I find that it's fairly clear where the players are headed in the next few sessions, so a setpiece in that time range is easier to prepare if you have one in mind. However never underestimate the power of failed rolls escalating to turn something seemingly trivial into an elaborate setpiece in its own right. Also once your players hit that grand location, you have no control over what they will do, so it doesn't get to play out like a cut scene and - I would argue it shouldn't. One thing that can be a big help in this situation is the ability to add custom moves for specific situations and environments - there is a knack to designing a move in the success/partial-success format that is fun and interesting, but there is a lot of good advice on it around so if your characters suddenly need to operate a complex magical cannon, pilot a mysterious ancient vehicle or whatever else you can imagine, you can create specific moves for them that empower them to respond but also guide them towards actions that might be more interesting and fit with the narrative.
• tightly woven plots - it is very hard in general to have a tightly woven plot without detracting from your players' agency, but if you know what the other agents in the world are doing then you can have some very taut plots running and they can also respond in interesting ways to the way the PCs behave. I find the idea of NPC agency and response probably more interesting than having a set of pre-wired plots that the player characters can slot into.
• The feeling of a complex world - this comes down to how much worldbuilding you do. With the game I am running now, the players weren't that invested in the worldbuilding so I ended up doing the majority of it so I have an outline of history and as I need to know more about a specific area I fill that in. However I do still use the approach of bringing their ideas into the game, but as ever once they have put something in the world, I own it as GM and will use it as I see fit.
• Being able to foreshadow future developments and provide shocking revelations This is very much what Dungeon World's Fronts are designed for - you have your dread portents that indicate the progress of a front and should foreshadow the future. Revelations are almost always a connection between some things that we already knew but which we hadn't connected - that can be easier if you have spaces in your background that allow you to build those connections.
• Present well-thought-out moral dilemmas - certainly Dungeon World favours the principle of offering the players hard choices and making those morally interesting rather than simple and immediate is purely a question of being aware of what dilemmas are available to you. A little like the setpieces ( or weather forecasts ) it's easier to predict what the players will do in the next couple of sessions and so try to have a few ideas on hand in case the opportunity comes up to use them. For a great example of using improvisational *World gameplay to create something of considerable substance I definitely recommend Friends At The Table which is a tremendous podcast.

So I think you can find the things you are looking for in this family of game, but if you want more control then maybe you should look for a game that offers you that by design- there are plenty of them around.

(As an aside, we're podcasting this game, so you can hear how it goes from the link on my profile, but bear in mind that we're basically all idiots, so although hilarity ensues it's probably the opposite of a textbook guide to running or playing Dungeon World. )

In common parlance, a 'Powered by the Apocalypse' game is a game that uses the same resolution engine of Apocalypse World, by Vincent D. Baker, or is somehow recognizably inspired by or similar to it.

In other word, it is a way for people to categorize Apocalypse World hacks.

Despite this being the perceived definition, commonly used in a large part of the Internet including the tag description in this site and Wikipedia, PbtA is in reality something different.

Vincent and Meguey Baker stated that:

"Powered by the Apocalypse" isn't the name of a category of games, a set of games' features, or the thrust of any games' design. It's the name of Meg's and my policy concerning others' use of our intellectual property and creative work.

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Here's the policy:

If you've created a game inspired by Apocalypse World, and would like to publish it, please do. If you're using our words, you need our permission, per copyright law. If you aren't using our words, you don't need our permission, although of course we'd love to hear from you. Instead, we consider it appropriate and sufficient for you to mention Apocalypse World in your thanks, notes, or credits section.

It's completely up to you whether you call your game "Powered by the Apocalypse." If you'd like to use our PbtA logo in your game's book design or trade dress, ask us, and we'll grant permission for you to do so. This isn't a requirement of any sort.

[...]

Again, "Powered by the Apocalypse" isn't the name of a kind of game, set of game elements, or even the core design thrust of a coherent movement. (Ha! This last, the least so.) Its use in a game's trade dress signifies ONLY that the game was inspired by Apocalypse World in a way that the designer considers significant, and that it follows our policy wrt others' use of our creative work.