What's the fastest way to transport mobs vertically in the Nether?

While the flying machine is certainly fast once built, and its pathway prepared (need margin so it won't stick to walls), probably the fastest way when considering time and effort put into preparation, is to prepare a clear, passable path (stairway) with no unforeseen obstacles, get a couple gold ingots and building blocks, then remove any golden armor you wear. This will aggro the piglins and they will start chasing you. Once you reach your destination, toss the ingots on the ground (or into whatever holes/cells you want the piglins to enter), hide somewhere, put your piece of golden armor on, and after about 30 seconds you'll be forgiven for your fashion faux-pas. You can then get the piglins exactly where you want them by throwing the golden ingots where you want them.

Alternatively, you may skip the whole (somewhat risky, especially in case of crossbow-wielding ones) chase scene and just lure the piglins with the ingots. You can also use other golden items if you have a surplus.

List of items you can use (after the Wiki ):

• Bell
• Block of gold
• Clock
• Enchanted golden apple
• Gilded blackstone
• Glistering melon slice
• Gold ingot (piglins barter something else in return)
• Gold ore
• Golden apple
• Golden axe
• Golden boots (worn after being examined)
• Golden carrot
• Golden chestplate (worn after being examined)
• Golden helmet (worn after being examined)
• Golden hoe
• Golden horse armor
• Golden leggings (worn after being examined)
• Golden pickaxe
• Golden shovel
• Golden sword (equipped if piglin is using a non-gold weapon or crossbow, or does not already have one)
• Light weighted pressure plate
• Nether gold ore

Golden nuggets also work, but piglins show very limited interest in chasing them down - they'll run quite a distance to reach an ingot, but only walk a couple blocks to get a nugget.

extra tip: a piglin with a crossbow will swap it for a golden sword if thrown one; this will help with keeping them chasing you instead of trying to snipe you from a distance.

Let me just add for people looking for transport down - push them down a shaft with a slime block on the bottom.