So in Dungeon World, everything breaks down to 'moves'. What about situations where a move is not available, but a check of some sort is called for, such as in the case of a non-fighter who wants to kick down the door? What mechanism do you use to resolve that situation? Is there the equivalent of a skill check?
No, there is no equivalent to a "skill check" in Dungeon World. Dungeon World operates on a different set of principles that don't require or really permit task-based resolution rolls. If you're playing DW, you have to give up the idea that everything requires a roll.
The most important principle for this question is that dice are only rolled when a move says to, and moves only happen when they are triggered. To trigger a move, a player has to describe their character taking a fictional action or set of actions that matches the move's trigger. Further, a character can't trigger a move that they don't have access to.
So what does it mean to try to break down a door? Forget about breaking down a door with the right move available—I mean, what does it mean to break down a door at all in DW?
This part right there is critical for DW to work. The players must describe not what they intend to accomplish, nor what move they want to trigger, but actually describe fictional actions that would trigger the move. Players don't ever get to say when a move triggers—that's the DM's job.
The way the system functions, how their actions are described that lead up to the move trigger is an integral part of what's happening in the game. A key part of how DW works is that the consequences of a miss can depend (possibly, depending on the move the DM chooses to respond with) directly on the fiction that the player created in order to trigger the move. If you skip that, you very quickly get lost both as players and DM and misses – and moves in general – get really confusing. If the player in the example above had instead described chopping at the door with their axe, bruising their shoulder would be a nonsensical miss result, right?
So, if you decide on a "right" move before you establish the triggering fiction, you can get into a situation like what you're running into with your question. So let's look at when you don't have the "right" move.
Or maybe instead:
Similarly, even moves that you might intuitively think should be exclusive don't need to prevent other classes from attempting the fictional activities they represent. The best example is the Wizard's Cast a Spell—only the Wizard can do that right?
Well, no. As a move it's limited, but actually, there's no reason that a non-wizard can't cast a spell, except lack of a gimme move that just says they can with easy-to-access fiction. If a Fighter wants to gather the Shards of Nemesil from the Nine Lost Vaults and consume a vial containing the whispered regrets of an orphan while chanting the Unmaking Dirge in order to summon the Apocalypse Dreadnought? They can totally try to do that, but they have to do it the hard way, building up the ritual within the actual game world without a nice, singular move to lean on. But if they don't get in too much trouble from the moves that the getting-ready-to-do-this-ritual fiction triggers along the way, even a Fighter can cast a spell. (Is it a good idea though? Heh, probably not. But that's the player's choice to make.)
So that's how you handle "not having a move" for what the player wants to do. Really, there is always a move, and you can find out the right move but simply asking the player to describe what they're doing to pursue their goal until they describe something that triggers a move they have access to. It might not be the move anyone would expect if you just picked the move first and then tried to figure out how to make it happen, but it will always be the right move, because the fiction has continuity and internal integrity.
In the case of a character wanting to take a move that is not available to his class, he can't take the move. Kicking down a door (Bend Bars/Lift Gates) is a simplistic example that it might at first glance seem should be available to everyone. But, take the move Cast A Spell. If a fighter decided that he wanted to cast a spell, it would be outside of the capabilities of the class. It's the same for kicking down a door. Dungeon World is made so that each character has his use, which lends itself towards the character having his chance in the limelight.
There are basic moves that cover a lot of things, and if what the character is attempting to do doesn't specifically fall in the purview of another class, those can be examined to see if something fits. But to give a character access to the same abilities that another class has is to water down the usefulness of the characters.
There are no skills, only moves.
If you need to get past a door and you don't have bend bars, lift gates, then you need to think about the situation differently. Say the room is filling with water and you need to escape. Your options could include defying danger to power through or discerning realities with What here is useful to me? to find a way out.
Here's a slightly different perspective:
The Bend Bars/Lift Gates move allows for you to avoid some of the effects attached to going through the door:
Reading the move, we see it could have one or more of the following effects: take a long time, damage valuables, make a lot of noise and/or completely destroy the door or bars or whatever.
So, if you start banging/hacking/chiseling at the door with your ax, hammer or mace, etc., perhaps using up some adventuring gear along the way, all of those could happen without ever triggering a move. From this perspective, the Move allows you to avoid some consequences, and going without the move doesn't. Then, with all that time + effort, the fiction will allow you to open the door, albeit with a bucketload of consequences.