I've only had one new baby fresh from the breeder, but I did put her right into her nice, new, big cage and she did fine.
I also debated all the way home with Pete: put him in his little cage he was used to, or right into my much bigger one. I kept going back and forth, and finally decided to just get everything new over with at once, and put him right into my big one. I think he was happy with it right off though. He climbed all over it, including across the top right away. I think he was happy to actually be able to move and flap his wings.
And of course Merlin had to move into a new cage right when she got here. I couldn't have her old one because it was a 2 story stacked cage and their amazon lived in the other half. He didn't go to his new owner until a week later. So KBEquine got that cage, but that's ok as she has adopted so many birds she probably needed it more than me. Merlin didn't seem to have any issue with her new cage.
Babies are also much more adaptable in general than older parrots who sometimes become very attached to their cage.