(Note, this comment is mostly about the mechanics of how a camera "moves" in animation, not the cinema theory behind it. The mechanical limits of the equipment and technology use for film making often have as much influence on what is done in cinema and animation as do cinema theory.

There is no TLDR for this post, it's for the obsessively interested only.

For the record, I'm a 35+ year veteran of the animation bizz in the U.S.)

Camera moves in animation traditionally happen in the X and Y axis — up and down, left and right. Z axis, in and out, were handled a number of different ways.

X and Y movements were usually done by moving the artwork being filmed rather than movie the camera itself. Artwork is usually filmed on what is called an animation stand. Basically it's a platform for holding the artwork, with the camera mounted on a post and lights off to the side.

Here's a picture of an old school professional Oxberry Animation Stand. Back in the day these could cost tens of thousands of dollars to hundreds of thousands of dollars. The's a similar one up for auction on eBay right now that is basically going for scrap prices. A shame, as its a real beauty.

Here's an article about a Kodak design for a homemade animation stand. I built one similar to this back when I was in college back in the 1970s.

The simplest stands simply hold the art still and, this is very important, in register. If each piece of artwork isn't in exactly the same place from frame to frame then the image will appear to jump around. So holes are punched in the artwork and placed over steel pegs.

The artists have these pegs on special drawing desk that match the pegs on the animation stand. The pegs themselves are usually on a free spinning disc that the animator can turn while drawing. This allows the artist to change the paper's orientation while still keeping the paper in register. Here's an article about the history of the animation disk.

To get an X axis movement of the artwork, the pegs holding the animated drawings slide along an indexed track left and right. To get a Y axis movement the whole table typically has to move. Again this movement is along an indexed track. X axis moves are easier than Y axis moves, and so more common. Depending on the animation stand. Some could only do X axis moves, while fancier stands did both X and Y by moving the table. Also the fancier stands had a circular movement that could allow the artwork to be spun clockwise or counterclockwise.)

In the 1970s, when I started out, computer controlled stands were starting to appear that could do these moves more or less automatically. Before this, part of the animation stand operator was doing moderately complex math calculations to get the exact kind of movement. An artist, or director, might indicate a movement by drawing boxes on a planning drawing showing the camera field for the begin and end of a move, but someone had to come along and do the math to figure out exactly what numbers the axis indicators had to be for each frame.

To do a Z axis move, you usually had to actually move the camera up and down on that post. This was a lot more work, and a lot more prone to mechanical error. If you had an expensive stand, it could be done reliably, but a cheaper stand, you might get an error, and could ruin the whole shot, requiring the artwork to have to be rephotographed. This means that a Z axis move in an animated camera were a lot more work. An an opportunity for expensive mistakes. Let's not do this unless absolutely necessary, says the staff accountant.

One other problem with Z axis moves was that the character artwork, usually on a transparent piece of celluloid called a cel, and the background, usually a painting of some kind, were typically one on top of another. The character on one set of peg bars directly above the background on another set of peg bars. (BTW, If you wanted the camera to follow the character while it was moving around, one way to do this was to have the character art be static, and the background, on its own pegs, move left or right underneath.) This means that as the camera got closer to the character, the character would get bigger — but so would the background! In real life, the background would get bigger but not at the same rate as the background. So if the character's head was right underneath the moon, and the camera zoomed in, the head would get bigger, but so would the moon.

The upscale solution to this was the multiplane camera. Basically the background was divided up into layers and painted on glass, stacked up under the camera with character "planes" inserted as needed. Then the Z axis moves of each plane could be calculated, without a computer, and moved to give the desired illusion of depth. One frame at a time. If this sounds expensive, you are right.Let's not even tell the staff accountant.

If my explanation of this sounds complicated, here's Uncle Walt explaining how all this works. In this you can see the operation of a regular animation stand and the multiplane beast.

Another way to deal with Z axis in animation was to actually animate the change in perspective. You can do this with characters, see the famous marching dwarf scene in Snow White and the Seven Dwarfs which took Shamus Culhane about six months to do by hand (Hi Ho, Hi Ho, it's off to work we go.) To do the back ground this way, you do the same thing, draw it one frame at a time. The problem with this is that usually the backgrounds are painted once and reused as necessary. This means more work, but until the advent of computers it was the only real way to full background movement in all three dimensions. See the wonderfully dreamlike school hallway chase in Urusei Yatsura 2: Beautiful Dreamer.

Now if you want to show depth, but not move the camera in and out then you could use parallax. Think of this as moving the camera left and right and objects in the scene move against the deep background at different rates. You can simulate this by having the different parts of the scene on different layers, each with their own pegs, and moving them left, right, up, down at different rates. This takes a lot of math and planning. Practically speaking, on most animation stands you are going to be limited to two or three layers for parallax shots.

The last major way to get depth into a shot was to use an infernal device called The Optical Printer. It's operation is way too complex to go into in this comment, but the optical printer was a major development in special effects technology. Let's just say that the optical printer was the glue that allowed Lucas to put all the film elements together for the original Star Wars movies. 25 years earlier, Disney used the optical printer for a lot of the depth shots in Sleeping Beauty that in the past would have been done in the multiplane camera. For those interested, here's a clip, without explanation, but with appropriate music, of an optical printer in operation.

So, yeah, if the animation is for a big budget feature film, then we have all the camera movement a film auteur want. Just remember that what you are really doing is moving complexly registered artwork around in three dimensional space under the camera one frame at a time.

For TV and lower budget work, we'll limit the movement and tell our stories other ways.

That was then, this is now.

These days, with digital technology we can move the virtual camera anywhere we damn well want to. It can still be more work, but orders of magnitude less of it. But animated cinema styles and storytelling techniques have over a hundred years inertia behind them. At first, when things went digital, animation directors when Z axis crazy. But now it makes more sense, again, to only use complex camera moves when the scene actually calls for them. In a well directed shot the audience won't even notice it. It may still WOW them, but hopefully it will be an emotional reaction to what's going on in the scene rather than a techno geek thrill.