How to transport animals for large distances?

If you want to transport them a long distance, I would recommend going up a few blocks, and building a long track with powered rails. Simply make it small (1 Block wide) and light it up so no mobs spawn.

I think using a track with a cart would work easily in any dimension or biome. I tried it out, but then again most of the time I play in creative. I'm not too experienced when it comes to survival because I can barely last a day. But I'm sure the track would be possible to survive on because if you use enough lights at night then mob can't spawn near it. Leads aren't the most reliable when it comes to animals because if you go to far away from the animals ( sprinting ) can cause the lead to break. If you are using lighting you could use torches, I used redstone lamps and daylight sensors. Mine is very long and if I can possibly make it through it to get to my house with an animals then any track could be possible. Though I wouldn't recommend going through desert biomes because most mobs are able to be there every night. If you do need to cross an ocean you can either make a path for your minecart over the water or use a boat because animals can ride with you. To protect myself from mobs other than endermen I made safe houses about every thousand blocks that are accesable for animals. I don't have much of a solution for travel in the nether except for making a tunnel so mobs can't attack you or getting weapons to attack mobs.

There are several ways you could achieve this:

1. Build a railway or canal: This will work all the time but it's quite a hard work (as you said, 20000 block away).
2. Nether travel + the first one. This will reduce the work but it's still quite a lot (20000 / 8 = 2500 meters long rail).
3. Cheat. If cheating is allowed and you don't bother doing so, you can use hacks like BoatFly to carry animals (any mobs and players are capable) and fly them to the distance. Remember, you can only carry one mob once, and boatfly could sometimes cause instant fallen death after leaving the boat even if there's no fall at all, or even with the NoFall hack, so be careful. Another important thing, BoatFly is not fast. Also works with nether acceleration.
4. Use lead: Slow but you do not need to build a channel. Also works with nether acceleration.

Hope it helps! Correct me if I'm wrong.

20,000 blocks? Nether roof is the most practical way.

You could use flying machines through overworld, through at that distance, they will be nearly guaranteed to glitch out eventually, plus if that's not in a straight line, you'll need to make an L turn somewhere along the way.

2,500 blocks of railway through the Nether is impractical, though doable. Probably an iceway for a boat would be more realistic, built a short distance under the Nether roof. Still, the time and effort to build the completely covered tunnel 2500 blocks long, while not excessive, is rather unpleasant.

Definitely the fastest and easiest way is to access the top of the Nether roof (there are many tutorials how to do it, e.g. like this ), build Nether portals near the start and end of the route, then just walk the animals the distance, using leads + whatever plant they like (like wheat) - that combination is least trouble.