In the past I'd started this process by encouraging them to approach and beak-target the harness voluntarily. This was to combat their behavior of flying away upon seeing the harnesses.
The next step of training was to get them very comfortable with the head loop. To do this, I started out by making a large loop out of the harness material and shaping the behavior of putting their heads through the loop.
Once they were comfortable with that, I gradually made the loop smaller and smaller until I could transition to using the actual head loop of the harness, which is very small (especially for a big, bulky caique head!).
Within five days, I had both birds (who had previously flown away from the harness, and occasionally bitten my partner when he tried to harness them) voluntarily seeking out the harness, finding the head loop, and putting their own heads through it! I was shocked at the rate of progress we made.
To illustrate the process, I made a video of some of my training steps:
(x-posted to parrots101, caiques, and clickertraining)