You haven't provided any specs whatsoever. Kinda important when talking about taking desktops on air travel. What/how big is this mATX case - is it a 19L Cerberus or a 41L Define Mini? You say it'll be checked luggage, so reason to need to take AIO off then, since you're not dealing with carry-on fluids restrictions (and even then it's not as much of an issue nowadays). Frankly an AIO is preferable to a gigantic air cooler for transportation purposes.
But really, if you value your PC you'll need to treat it like a company would ship a prebuilt, and take the same precautions. Take heavy triple-slot GPUs out, or use those expanding InstaPak padding / "airbags" to ensure that all the empty space inside the case is filled. I'd package it like a prebuilt as well. Do all of the above precautions to protect the inside of the computer, then use the mATX case's original box with the original styrofoam packaging, fill it to make sure there's no empty space/wiggle room inside the cardboard box.
If this case is as big as I'm guessing it is, I
really wouldn't just throw it in a suitcase and wrap it with clothes and towels. As a carry-on? Sure. As checked baggage? That's not going to make an ounce of difference in protection unless there's a 1-foot thick layer of towels in there.
At least on the SFF scene, having to check your PC in as checked luggage,
without proper packaging, whether intentionally or unintentionally (when you don't plan ahead and it doesn't qualify as a carry-on) basically means to expect your PC to be damaged. There are things that baggage handlers do, that not even a properly kitted out Pelican case can save a computer from.
As for a monitor as large as a 27", I'd do the same - fully packaged into the original box and checked in as such. Smaller monitor you could conceivably throw into a bag - 27" is on the large side for that.
Communicate to the baggage/check-in agent what is in the box and the importance of handling it with care. If you throw either the PC or monitor into a suitcase, this will not be something they can help you with. Should be no issue with the proper markings, heads up to airline employees, and fragile stickers, as it'll be clearly identifiable as a monitor from the box, just as properly packaging the PC itself would allow employees and handlers to clearly recognize what's in the box.
So im planning to move out in a couple of weeks and i have a 27inch monitor with a matx pc that i'm bringing with me. My old build was a matx case as well and i transported it safely over the plane as well but that was with a ryzen stock cooler. So now i'm using a 240mm aio id cooling zoomflow 240x. Do i need to take off the aio? I didnt take off the ryzen stock cooler back then. I'm planning to use the original pc box with everything in it except the gpu. I'm going to put the gpu back to its original box. I have no hdd as well so i think i should be okay. As for the monitor, i have no idea how do i transport the money. I brought back a 24inch monitor before but that was because it could fit in my luggage this 27 inch cant. Oh yeah i forgot to mention, these will be checked in luggage so no hand carry.