The process of introducing new additions to a cage/aviary takes some time and has to be done correctly for good results. Any new birds you plan on introducing to a cage will have to go through a period of quarantine. They will need to be placed in a separate cage, but placed near their soon-to-be cage so that the other birds can get used to them and you can observe for any illnesses in the new bird. If all is well after the first week, you should be able to bring the cages right up to one-another so that the finches can meet face-to-face.
The first physical contact should take place in the area of the cage where the birds are likely to spend a lot of time together, be it a perch or a feeding station. If this area is another bird’s territory, rearrange the fixtures and fittings to give the feel of a new, neutral space. Food stations should be placed on both sides of the cage so that the birds can feed separately.
New birds will need to be caged separately for a while before being allowed to live together
Bickering and some flapping around is to be expected and is nothing you should worry about. You should not intervene unless you notice persistent violence. The birds will calm down once the hierarchy has been established. You can offer your birds some of their favorite food or a treat to take their minds off the confrontation via group feeding.
If you notice that after a day or two the birds are still fighting, you may have to consider separating them again for a longer period of time. The larger the cage, the easier the transition should be.