How do I transport fragile items?

The original packaging boxes are a good place to begin; your monitor survived its journey from the factory to your house in that box. Your main concern will be about protecting the box from other items falling onto the box, which you can mitigate by storing the monitor in a hard-shell case if possible. (I don't bother with this for wine.)

It is also worth noting that under the Montréal Convention, an airline participating in an international journey is strictly liable for damage to luggage in its care up to about 1500 USD, regardless of any contractual term or waiver. I won't go into the legal detail about that but it provides you with a measure of insurance in the unlikely event of damage.

I am regularly shipping bottles of wine around in polystyrene boxes as luggage on aircraft; no damage yet. Personally I would say pack your monitor as well as you can, try to protect the screen side from impact damage, and ask the airline to tag it as fragile. It will probably survive > 95% of journeys.

The best it to have the original box. This usually comes with styrofoam that fit the shape exactly of the monitor. Within it the monitor is inside a large anti-static bag and I plastic wrap the box for protection in case it gets wet.

As thing would have it, this year I travelled twice with 2 monitors. One was in its original box since still had it. Nothing happened to it. It was labelled on 3 sides over the plastic and 3 sides under the plastic in case it got unwrapped. The airline also affixed fragile labels on 3 sides and had someone carry it to the transport rather than going on the conveyor belt. It survived perfectly 4 flights this way.

The other monitor was an older but still high-end color-calibratable model for which I no longer had the original box. That one went into a hard-shell Samsonite (other brands make them too but I own four of these and they outlasted newer ones I bought later) suitcase. The monitor was first wrapped in bubble wrap (the kind with large, roughly 1" bubbles) and then wrapped in towels. Another time I had it inside a rolled yoga-mat. The monitor arrived intact from all its flights. Make sure the suitcase is full of softish items though and do not leave empty space as that will cause the monitor to shift and it may lose a bit of its protection.

Anecdotally, at one of the stopovers I was asked by customs to come watch them inspect my luggage and they kept asking why I would be travelling with a monitor! I had to explain them it was special and showed colors that others didn't.

Wrapping it massively in bubble wrap, and asking the airline to put a fragile sticker can help. I carry bottles of alcohol on almost every flight, and have had zero breakage so far. In close to twenty years. So it's mostly down to your wrapping - and the airline of course...

An original box will only survive a few times. They are designed to ship to the end user once and maybe for return to the manufacture.

The proper option is a hard case with foam inside cut to fit the monitor. These cases are designed for transporting expensive and delicate equipment. The are water proof and designed to survive significant abuse.

https://en.wikipedia.org/wiki/Transit_case