Woah. That is alarming looking. Budgies are originally from Australia which also tends to be hot. Assuming your method of transport has air conditioning, I'd take them to an avian certified vet ASAP. If you don't already have a regular CAV, we have a resource list now put together here:
After a suggestion by a member, @Abu Colo, the Mod Team decided to create a list of veterinarians around the world who see birds. This was something we had discussed a few years ago, but the task seemed so large that we didn't get very far. However, the need is there and we decided that the time...
www.parrotforums.com
I know budgies, like Amazons, can get fatty tumors (called lipomas), and that might be what this is. Those tend to be benign and will sometimes go away if you correct their diet. However, it could just as easily be something more serious, and it would be best to have them seen by an experienced veterinarian. Even benign lipomas can cause problems with agility and flight if they grow too large. Normally they're a yellowish color - the photo of her from the front shows a reddish area that concerns me a bit as that seems to indicate irritation or inflammation. Your vet will be able to diagnose this and form a treatment plan.
