There are a lot of problems with this affinity. More probably than any other (have tried the majority, but not all). Hopefully we can get them fixed for a more balanced and enjoyable game.

Under One Banner

Their affinity action is called Under One Banner. It lets you annex outposts or administrative centres by parking an army on that tile and paying a sum of gold (you won't know how much till you're parked there) then wait a certain amount of turns (again, you aren't told how long till you do it). Here are the problems:

1) if you move other units into that army, it stops the annexing and you have to do the action again, paying the sum of gold again and waiting the full set of turns again. I think that neither of these things should happen, it should just carry on with the annexing. Especially since it doesn't tell you that this will happen, it feels unfair to punish you like this. Also, for consistency, ransacking works where it just continues when you reinforce the ransacking army, so this should, too.

2) If you move the army off, it stops the annexing and you have to do the action again, paying the sum of gold again and waiting the full set of turns again. It does this even if you move off, then back on in the same turn.

3) If they wish to keep it the way it is for 1 and 2, they should give you a popup warning before you do any of these things, saying "Doing this will stop the annex. Are you sure you wish to do this." and give you two options as buttons to click, to proceed or cancel.

4) Under One Banner can't be used when you're at war with that country, but there is nothing in the game that tells you this. So, you could choose an expansionist civ not knowing this, and get punished hard for it. Happened to me and probably will happen to everyone.

5) You should be told beforehand the price in turns and gold. Otherwise you can travel all the way to somewhere just to find out you can't pay. Or it'll take too long and you're enemy will surely reach and destroy you by then.Or jsut tell us how it's calculated. Tell us roughly what to expect (like why one spot will take 4 turns while another will take 6, or one will cost 150 and another 300) We need to know why some places are more expensive than others so we know what to target. It wasn't obvious in my game. A place I annexed in the ancient era took longer and costed more than one in the classical and they both were pretty comparable (both no districts, low pop, the ancient had 1 lux, the other 0)

Expansionist Era Stars 

Expansionist Affinity gives you bonus fame for Expansionist Era Stars. The problem is, they are by far the hardest stars to get, even when you try hard to get them! Right now I'm playing a game where I chose 2 Expansionist civs for the first 2 eras (Assyrians and Persians). I had a good amount of space and only 1 immediate neighbour who was quite weak (in ancient era he soon became last place and stayed there). In Ancient era I warred with him, took all his territories except his capital, took all the remaining empty territories up to my farther neighbour and the neutral states. I focused heavily on influence production. I built emblematic districts in every single territory and maxed out on liberty. I forewent building wonders. I attached every outpost, attaching only to adjacent ones for the cheapest rate.  And with all that, I only got 1 expansionist era star in the ancient era...and it was my last one!....and it was my 13th star!!! I literally had to wait 6 extra stars to progress in order to get the star I specialize in, that's so ridiculous!!! And the Classical era has turned out no different. I'm at 7 stars, turn 88. The only two categories I don't have are expansionist and merchant. I'm at 12/14 for expansionist. There are no empty territories on the map I've explored so far. This era I've Assimilated all 3 of  the independent people I've seen. I'm trying to use Under one Banner on my neighbour to the north but it's so slow going (I just got out of a war with him, and immediately went to do this).

So Expansionist Stars need to be buffed significantly. At least lower it by 2 in each era. Probably needs to be even more than that though, honestly. I wasn't even trying for the other stars, and was trying hard for expansionist and expansionist came in dead last.

Ambition

Expansionist's Ambition ability to always trespass has many bugs (which honestly nerf it so badly). I've detailed these in a bug report I just made: Issues with Expansionist's Ambition ability to always trespass

Please share your thoughts, and thanks a lot for reading!

You said it elaboratedly what I have said before. I play on Endless and Huge and I need 20 Territories! With the new costs? It is ridiculous. An abhorrent amount of work is needed to get even 1 star and it is not hat OP in itself, because once you get a territory, you still have to build on them. Influence is very hard to get. Make it worthwhile please. Also, Aesthate Stars are very hard also, even with an influence-heavy nation. And they are somewhat Co-related. People rampage on with Food and Industry and they can do everything. I saw a person who made 1200! (you read right 1200!) population on turn 300 of Endless. Just because there are so many exploits. And if you are an expansionist, you struggle, because you can't get enough stars to even compete. As OP said, even the first star is a pain in the ***. You know, others are amassing Agrarian, Builder and Scientist stars. And, if you are with many neighbours around you, you can't even take their territories without the others doing something about it, especially when they are in alliances. That's why the 18-starred Expansionist Achievement is done by 0% of people on Steam. Help us please, by reducing the number of territories required to acquire. With the new, upscaled costs it is only reasonable, if done so.

Thanks for sharing your experience, Balashka! Wow, 20 territories, that is absolutely ridiculous (and I thought it was bad on standard speed and normal size). I can't imagine ever getting the third expansionist star, it's just not at all feasible (in any size or speed). I agree with all your points. Hopefully more people complain about this so the devs take note and balance it, as well as fix the other issues I mentioned in the original post.

Maybe expansionist affinity should have a much cheaper influence cost to attach territories?