I've traveled a lot with a Brompton, but I'm not a fan of having to travel with a box for the Brompton (ie I don't want to carry it on a bicycle tour). So I make a box whenever I need one, and the only things I travel-with to make this box are:
- About 10 meters of paracord
- A knife
- A permanent marker

Some days before I fly, I'll also get the following:
- A roll of plastic packaging tape
- 3-4 medium-sized boxes of cardboard (or big, if possible)
Luggage Fees
First, you do want to select your airline carefully. The Brompton is small enough to pack-up under the oversize luggage fees charged when shipping normal/big bikes on a plane, but weight may be an issue.
Brompton says their bike weighs 9-13 kg. With my upgrades, accessories, and added padding for shipping, my Brompton usually comes-to 20 kg. YMMV, but -- personally -- I avoid airlines that limit checked bags to max 18 kg (though 23 kg or 30 kg is a more common limit).
Obtaining Materials
I buy a whole new roll of packaging tape. I use about half of the roll, and I always travel with the last half, just in-case it the box needs to be opened/re-sealed.
I've never bought cardboard. I've packed my brompton for flying in North America, South America, Asia, and Europe. In every country I found cardboard for free and recycled it.
Preparing Brompton
The first thing I do is prepare the Brompton, so it's well protected.
The Brompton is a pretty rugged beast, and it's protects itself pretty well when folded, but I'm particularly interested in protecting the following vulnerable/expensive-to-replace parts:
- The hinges
- The front mounting block
- The wheel spokes
- The pedals
First, I unscrew the bolts and clamps from the hinges. I put these in a ziplock baggie and I either check it with my other luggage or wrap it in cardboard and tape it inside the folded brompton's triangle or something.
Second, I rip/cut pieces cardboard from the smallest box I have, and I bend the strips around the above-listed parts to add padding to protect them (this will, of course, be redundant padding in-addition to the box you'll put it in).

Don't be afraid to use a lot of cardboard and tape! You'll be making your Brompton wider, longer, and taller in dimensions, but the Brompton is so small that you're very unlikely to pad the brompton so much that you get slapped with a oversize luggage fee (generally max 158 cm).

Finally, when I'm satisfied Brompton's vulnerable components are well-covered in cardboard, I wrap it in a sleeping pad. And if I'm traveling with a big (synthetic) winter jacket, I'll cover it with that too
Making the Box
When the Brompton is sufficiently covered in padding, it probably won't fit in any of the cardboard boxes. So I usually open one side of the biggest box until the Brompton fits in it, and then bend a new edge into the cardboard so it fits perfectly.
I may re-bend other walls of this big box so that the box fits the Brompton perfectly.
Then I'll use the remnants of the other couple boxes to build-out a top of the box and reinforce any sides where one sheet of cardboard meets the other. Use just a small bit of tape as you go.
When the box is completed around the Brompton, I write my name and every airport code the box should travel to/through on literally every side of the box. Maybe this is paranoid, but I did arrive to an airport one time and my Brompton did not. Fortunately, it arrived within a couple days. I'm glad it was clearly marked and not lost; checked luggage with this airline was only insured up to $3,000.
After clearly writing your info on the box, tape it up really, really, really well (you want to write the info before this because you can't write on plastic tape). If you built a box by combining multiple boxes together, I can't stress how important it is to use a lot of tape.

Tie it up
The resulting ~20kg box should offer great protection of your precious Brompton, but it'll be large, heavy, and clumsy to carry.
To make this easier, I usually just wrap & tie about 10 meters of paracord around the box, so it's easier for me and the airline workers handle it.

Bonus: The result doesn't look like the contents are very valuable, and I've never had an airline ask me what's inside; the last thing they'd expect is an expensive bicycle.
Alternatively: Suitcase
The first time I shipped a Brompton, I bought a secondhand Samsonite suitcase that was 158 cm (L+W+H).

I still prepared it with the cardboard on the vulnerable parts and it just barely fit inside the suitcase.
This was the last time I shipped the brompton this way, because I didn't want to carry a suitcase with me on a bicycle tour through the desert. But it could be a good option if you're not cycle touring.