I got Casey nearly 12 years ago. She went from the breeders to a mutual friend of the family's home (for one evening - they were expecting me there, but I was home busy doing school stuff) and then to me. I don't know who this breeder is or was, but she traveled over 30 miles before she was mine.
A few months after her, I got my bourke parakeet. She was 5-7 years old at the time and had one owner prior to me. She went over 30 miles to get to my home as well.
Ten years ago this January, I moved about 15 miles from where I lived then. (moving back in the direction they came from!) I had both of them as well as some other birds.
Pistachio came to me in 2005. In 1998, he was picked up from an animal shelter in California. I've only kept birds for 13 years, and had known Pistachio before I got my own bird. (not the reason I got into birds!) A family had recently moved into my neighborhood and had him and some other birds, so he had already been moved at least twice. The family actually ended up moving again, before I moved myself. About a year after I moved (say 2004 maybe?) they asked me if I wanted their birds. I said yes (several times, as they asked me several times), but they never brought them over, so I didn't actually expect them.... until one morning they were in my driveway and I didn't even know the family was bringing the birds over!
Early 2006 I took in another cockatiel, Tomi Girl. She was a bit of a rescue, considering she was nearly starved to death. I had known about her for several months, maybe a year, but she wasn't in trouble until the day I saw that she was nearly starved to death. All I know about her is that the people who had her before me found her and decided to keep her without looking for her previous home. (IMO, these people should not exist...)
Late 2006 I ended up adopting Charlie and had to travel 200 miles one way to get him. He was 12 years old at the time, was probably born around the San Francisco/Santa Rosa area, had one owner for 10 years, then lived in a pet store for a year then in a foster home for a year.
Late 2007, I was paid for a "service" with a cockatiel, Faye. She was supposed to be given to me but the owners decided to keep her... About a year later, I house sat for them, and she was my payment. As far as I know, I am her 4th owner. Her first gave her away at a garage sale (assuming that was her first home). I don't know how long the second people had her for, but they gave her to third home as a bit of a "house warming" gift (they didn't move far... nor did Faye between 2nd and 3rd home - and then to me - just up and down a street) and they had two other tiels at the time.
2011 came around and I had to move yet again... about 17 miles and closer to where Casey and the bourke came from.
Nearly 4 months ago, I welcomed Jayde into my home. She had one home for less than 3 years before going back to her breeders home, then into a foster home for 6 months. She then flew from Ohio to Nevada. That was a somewhat stressful event! Found out she didn't leave the last airport she was at until *AFTER* she was supposed to have arrived...
From my experience, as long as your birds are ok with some amount of change, moving is not that big of a deal. Yes, it can be upsetting to them as they get accustomed to a new place and different cage arrangement, but they'll settle in! Jayde has been having a bit of a hard time fully settling in because she's such an insecure bird, although she's doing far better now than she did previously. She was very scared and couldn't be handled at first. Charlie wanted nothing to do with me in his foster home but as soon as I got him home he was instantly clinging to me! More so I'm sure because of being in a new and scary environment rather than wanting to be with me... he just needed to be close to someone to know that everything was ok.
I've taken Charlie and Casey places. Casey handled it well back in the day, but now she has a flock she doesn't like being separated from. Charlie is rather loud at first when we go places but settles down once we get there and isn't so bad on the way back. I've only taken Jayde out once, but that should change soon enough!